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id="toc-Non-normed_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-normed_spaces"> <div class="vector-toc-text"> 1.2 Non-normed spaces </div> </a> <ul id="toc-Non-normed_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> 2 Definition </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition subsection </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Defining_topologies_using_neighborhoods_of_the_origin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Defining_topologies_using_neighborhoods_of_the_origin"> <div class="vector-toc-text"> 2.1 Defining topologies using neighborhoods of the origin </div> </a> <ul id="toc-Defining_topologies_using_neighborhoods_of_the_origin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Defining_topologies_using_strings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Defining_topologies_using_strings"> <div class="vector-toc-text"> 2.2 Defining topologies using strings </div> </a> <ul id="toc-Defining_topologies_using_strings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Topological_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Topological_structure"> <div class="vector-toc-text"> 3 Topological structure </div> </a> <button aria-controls="toc-Topological_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Topological structure subsection </button> <ul id="toc-Topological_structure-sublist" class="vector-toc-list"> <li id="toc-Invariance_of_vector_topologies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Invariance_of_vector_topologies"> <div class="vector-toc-text"> 3.1 Invariance of vector topologies </div> </a> <ul id="toc-Invariance_of_vector_topologies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local_notions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Local_notions"> <div class="vector-toc-text"> 3.2 Local notions </div> </a> <ul id="toc-Local_notions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metrizability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metrizability"> <div class="vector-toc-text"> 3.3 Metrizability </div> </a> <ul id="toc-Metrizability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Completeness_and_uniform_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Completeness_and_uniform_structure"> <div class="vector-toc-text"> 3.4 Completeness and uniform structure </div> </a> <ul id="toc-Completeness_and_uniform_structure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 4 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Finest_and_coarsest_vector_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finest_and_coarsest_vector_topology"> <div class="vector-toc-text"> 4.1 Finest and coarsest vector topology </div> </a> <ul id="toc-Finest_and_coarsest_vector_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cartesian_products" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartesian_products"> <div class="vector-toc-text"> 4.2 Cartesian products </div> </a> <ul id="toc-Cartesian_products-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite-dimensional_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite-dimensional_spaces"> <div class="vector-toc-text"> 4.3 Finite-dimensional spaces </div> </a> <ul id="toc-Finite-dimensional_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-vector_topologies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-vector_topologies"> <div class="vector-toc-text"> 4.4 Non-vector topologies </div> </a> <ul id="toc-Non-vector_topologies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Linear_maps" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_maps"> <div class="vector-toc-text"> 5 Linear maps </div> </a> <ul id="toc-Linear_maps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> 6 Types </div> </a> <ul id="toc-Types-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dual_space" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dual_space"> <div class="vector-toc-text"> 7 Dual space </div> </a> <ul id="toc-Dual_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 8 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Neighborhoods_and_open_sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Neighborhoods_and_open_sets"> <div class="vector-toc-text"> 8.1 Neighborhoods and open sets </div> </a> <ul id="toc-Neighborhoods_and_open_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Hausdorff_spaces_and_the_closure_of_the_origin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-Hausdorff_spaces_and_the_closure_of_the_origin"> <div class="vector-toc-text"> 8.2 Non-Hausdorff spaces and the closure of the origin </div> </a> <ul id="toc-Non-Hausdorff_spaces_and_the_closure_of_the_origin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_and_compact_sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_and_compact_sets"> <div class="vector-toc-text"> 8.3 Closed and compact sets </div> </a> <ul id="toc-Closed_and_compact_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_properties"> <div class="vector-toc-text"> 8.4 Other properties </div> </a> <ul id="toc-Other_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_preserved_by_set_operators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_preserved_by_set_operators"> <div class="vector-toc-text"> 8.5 Properties preserved by set operators </div> </a> <ul id="toc-Properties_preserved_by_set_operators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 10 Notes </div> </a> <button aria-controls="toc-Notes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Notes subsection </button> <ul id="toc-Notes-sublist" class="vector-toc-list"> <li id="toc-Proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proofs"> <div class="vector-toc-text"> 10.1 Proofs </div> </a> <ul id="toc-Proofs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 11 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 12 Bibliography </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 13 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Topological vector space</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 23 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-23" aria-hidden="true" > 23 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D9%85%D8%AA%D8%AC%D9%87%D9%8A_%D8%B7%D9%88%D8%A8%D9%88%D9%84%D9%88%D8%AC%D9%8A" title="فضاء متجهي طوبولوجي – Arabic" lang="ar" hreflang="ar" data-title="فضاء متجهي طوبولوجي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_vectorial_topol%C3%B2gic" title="Espai vectorial topològic – Catalan" lang="ca" hreflang="ca" data-title="Espai vectorial topològic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Topologick%C3%BD_vektorov%C3%BD_prostor" title="Topologický vektorový prostor – Czech" lang="cs" hreflang="cs" data-title="Topologický vektorový prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Topologischer_Vektorraum" title="Topologischer Vektorraum – German" lang="de" hreflang="de" data-title="Topologischer Vektorraum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Topoloogiline_vektorruum" title="Topoloogiline vektorruum – Estonian" lang="et" hreflang="et" data-title="Topoloogiline vektorruum" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_vectorial_topol%C3%B3gico" title="Espacio vectorial topológico – Spanish" lang="es" hreflang="es" data-title="Espacio vectorial topológico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Topologia_vektora_spaco" title="Topologia vektora spaco – Esperanto" lang="eo" hreflang="eo" data-title="Topologia vektora spaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D8%A8%D8%B1%D8%AF%D8%A7%D8%B1%DB%8C_%D8%AA%D9%88%D9%BE%D9%88%D9%84%D9%88%DA%98%DB%8C%DA%A9%DB%8C" title="فضای برداری توپولوژیکی – Persian" lang="fa" hreflang="fa" data-title="فضای برداری توپولوژیکی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_vectoriel_topologique" title="Espace vectoriel topologique – French" lang="fr" hreflang="fr" data-title="Espace vectoriel topologique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%84%EC%83%81_%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="위상 벡터 공간 – Korean" lang="ko" hreflang="ko" data-title="위상 벡터 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_vektor_topologis" title="Ruang vektor topologis – Indonesian" lang="id" hreflang="id" data-title="Ruang vektor topologis" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_vettoriale_topologico" title="Spazio vettoriale topologico – Italian" lang="it" hreflang="it" data-title="Spazio vettoriale topologico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99_%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99" title="מרחב וקטורי טופולוגי – Hebrew" lang="he" hreflang="he" data-title="מרחב וקטורי טופולוגי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Topologische_vectorruimte" title="Topologische vectorruimte – Dutch" lang="nl" hreflang="nl" data-title="Topologische vectorruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E4%BD%8D%E7%9B%B8%E7%A9%BA%E9%96%93" title="線型位相空間 – Japanese" lang="ja" hreflang="ja" data-title="線型位相空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Spassi_vetorial_topol%C3%B2gich" title="Spassi vetorial topològich – Piedmontese" lang="pms" hreflang="pms" data-title="Spassi vetorial topològich" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_liniowo-topologiczna" title="Przestrzeń liniowo-topologiczna – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń liniowo-topologiczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_vectorial_topol%C3%B3gico" title="Espaço vectorial topológico – Portuguese" lang="pt" hreflang="pt" data-title="Espaço vectorial topológico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Топологическое векторное пространство – Russian" lang="ru" hreflang="ru" data-title="Топологическое векторное пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Topologiskt_vektorrum" title="Topologiskt vektorrum – Swedish" lang="sv" hreflang="sv" data-title="Topologiskt vektorrum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D1%96%D1%87%D0%BD%D0%B8%D0%B9_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Топологічний векторний простір – Ukrainian" lang="uk" hreflang="uk" data-title="Топологічний векторний простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%8B%93%E6%92%B2%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="拓撲向量空間 – Cantonese" lang="yue" hreflang="yue" data-title="拓撲向量空間" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8B%93%E6%92%B2%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="拓撲向量空間 – Chinese" lang="zh" hreflang="zh" data-title="拓撲向量空間" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1455249#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Topological_vector_space" title="View the content page [c]" 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vector spaces">Topological vector spaces</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Vector space with a notion of nearness</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. A topological vector space is a <a href="/wiki/Vector_space" title="Vector space">vector space</a> that is also a <a href="/wiki/Topological_space" title="Topological space">topological space</a> with the property that the vector space operations (vector addition and scalar multiplication) are also <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a>. Such a topology is called a vector topology and every topological vector space has a <a href="/wiki/Uniform_space" title="Uniform space">uniform topological structure</a>, allowing a notion of <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a> and <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">completeness</a>. Some authors also require that the space is a <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> (although this article does not). One of the most widely studied categories of TVSs are <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex topological vector spaces</a>. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>, <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> and <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a>. Many topological vector spaces are spaces of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, or <a href="/wiki/Linear_map" title="Linear map">linear operators</a> acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of <a href="/wiki/Limit_(mathematics)#Function_space" title="Limit (mathematics)">convergence</a> of sequences of functions. In this article, the <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar</a> field of a topological vector space will be assumed to be either the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> or the <a href="/wiki/Real_number" title="Real number">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}"> unless clearly stated otherwise. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=1" title="Edit section: Motivation">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Normed_spaces">Normed spaces</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=2" title="Edit section: Normed spaces">edit</a>]</div> Every <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> has a natural <a href="/wiki/Normed_vector_space#Topological_structure" title="Normed vector space">topological structure</a>: the norm induces a <a href="/wiki/Metric_space" title="Metric space">metric</a> and the metric induces a topology. This is a topological vector space because[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]: <ol><li>The vector addition map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thickmathspace" /> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a228b45d8126790d6a6c15de38201617f188c61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.885ex; height:2.343ex;" alt="{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\mapsto x+y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto x+y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5743e1f4e664dfb03d15296905ee8710bccaaee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.268ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto x+y}"> is (jointly) continuous with respect to this topology. This follows directly from the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> obeyed by the norm.</li> <li>The scalar multiplication map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot :\mathbb {K} \times X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot :\mathbb {K} \times X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a435ad50a2d001bb82f609cc6d5cd2418bb6066e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.807ex; height:2.176ex;" alt="{\displaystyle \cdot :\mathbb {K} \times X\to X}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s,x)\mapsto s\cdot x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>s</mi> <mo>⋅</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s,x)\mapsto s\cdot x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a1234e302876072761f0701042d5385e0cd453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.624ex; height:2.843ex;" alt="{\displaystyle (s,x)\mapsto s\cdot x,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> is the underlying scalar field of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm.</li></ol> Thus all <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a> and <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> are examples of topological vector spaces. <div class="mw-heading mw-heading3"><h3 id="Non-normed_spaces">Non-normed spaces</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=3" title="Edit section: Non-normed spaces">edit</a>]</div> There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> on an open domain, spaces of <a href="/wiki/Infinitely_differentiable_function" class="mw-redirect" title="Infinitely differentiable function">infinitely differentiable functions</a>, the <a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz spaces</a>, and spaces of <a href="/wiki/Test_function" class="mw-redirect" title="Test function">test functions</a> and the spaces of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> on them.<a href="#cite_note-FOOTNOTERudin19914-5_§1.3-1">[1]</a> These are all examples of <a href="/wiki/Montel_space" title="Montel space">Montel spaces</a>. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by <a href="/wiki/Kolmogorov%27s_normability_criterion" title="Kolmogorov's normability criterion">Kolmogorov's normability criterion</a>. A <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological field</a> is a topological vector space over each of its <a href="/wiki/Field_extension" title="Field extension">subfields</a>. <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=4" title="Edit section: Definition">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Topological_vector_space_illust.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Topological_vector_space_illust.svg/220px-Topological_vector_space_illust.svg.png" decoding="async" width="220" height="265" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Topological_vector_space_illust.svg/330px-Topological_vector_space_illust.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Topological_vector_space_illust.svg/440px-Topological_vector_space_illust.svg.png 2x" data-file-width="386" data-file-height="465" /></a><figcaption>A family of neighborhoods of the origin with the above two properties determines uniquely a topological vector space. The system of neighborhoods of any other point in the vector space is obtained by <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a>.</figcaption></figure> A topological vector space (TVS) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over a <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> (most often the <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> numbers with their standard topologies) that is endowed with a <a href="/wiki/Topological_space" title="Topological space">topology</a> such that vector addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thickmathspace" /> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a228b45d8126790d6a6c15de38201617f188c61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.885ex; height:2.343ex;" alt="{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}"> and scalar multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot :\mathbb {K} \times X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot :\mathbb {K} \times X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a435ad50a2d001bb82f609cc6d5cd2418bb6066e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.807ex; height:2.176ex;" alt="{\displaystyle \cdot :\mathbb {K} \times X\to X}"> are <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous functions</a> (where the domains of these functions are endowed with <a href="/wiki/Product_topology" title="Product topology">product topologies</a>). Such a topology is called a <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>vector topology or a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">TVS topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> Every topological vector space is also a commutative <a href="/wiki/Topological_group" title="Topological group">topological group</a> under addition. Hausdorff assumption Many authors (for example, <a href="/wiki/Walter_Rudin" title="Walter Rudin">Walter Rudin</a>), but not this page, require the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> to be <a href="/wiki/T1_space" title="T1 space">T1</a>; it then follows that the space is <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, and even <a href="/wiki/Tychonoff_space" title="Tychonoff space">Tychonoff</a>. A topological vector space is said to be <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">separated if it is Hausdorff; importantly, "separated" does not mean <a href="/wiki/Separable_space" title="Separable space">separable</a>. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed <a href="#Types">below</a>. Category and morphisms The <a href="/wiki/Category_(category_theory)" class="mw-redirect" title="Category (category theory)">category</a> of topological vector spaces over a given topological field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> is commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {TVS} _{\mathbb {K} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {TVS} _{\mathbb {K} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0e22e4b7f9a6fb0fcc9609c6c8d1ae85bf117f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.224ex; height:2.509ex;" alt="{\displaystyle \mathrm {TVS} _{\mathbb {K} }}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {TVect} _{\mathbb {K} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {TVect} _{\mathbb {K} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/138e3277d6c4a816a1de4dfae7919d3a4e1dccc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.548ex; height:2.509ex;" alt="{\displaystyle \mathrm {TVect} _{\mathbb {K} }.}"> The <a href="/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a> are the topological vector spaces over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> and the <a href="/wiki/Morphism" title="Morphism">morphisms</a> are the <a href="/wiki/Continuous_linear_map" class="mw-redirect" title="Continuous linear map">continuous <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }">-linear maps</a> from one object to another. A <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">topological vector space homomorphism (abbreviated <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">TVS homomorphism), also called a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><a href="/wiki/Topological_homomorphism" title="Topological homomorphism">topological homomorphism</a>,<a href="#cite_note-FOOTNOTEKöthe198391-2">[2]</a><a href="#cite_note-FOOTNOTESchaeferWolff199974–78-3">[3]</a> is a <a href="/wiki/Continuous_map" class="mw-redirect" title="Continuous map">continuous</a> <a href="/wiki/Linear_map" title="Linear map">linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71ed0cbbb508da53826d7ea1f288394da3664937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.634ex; height:2.176ex;" alt="{\displaystyle u:X\to Y}"> between topological vector spaces (TVSs) such that the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:X\to \operatorname {Im} u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Im</mi> <mo>⁡</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:X\to \operatorname {Im} u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49991ad4ac7c2b5b9c9af2d3944bbd999d98c072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.353ex; height:2.176ex;" alt="{\displaystyle u:X\to \operatorname {Im} u}"> is an <a href="/wiki/Open_mapping" class="mw-redirect" title="Open mapping">open mapping</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} u:=u(X),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mi>u</mi> <mo>:=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} u:=u(X),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c6a7e11d8102d33226ea92540f9e830a303fc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.003ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} u:=u(X),}"> which is the range or image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30dcc93e14b40416ed2d1391bc6c08ee99fa5ff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle u,}"> is given the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}"> A <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">topological vector space embedding (abbreviated <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">TVS embedding), also called a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">topological <a href="/wiki/Monomorphism" title="Monomorphism">monomorphism</a>, is an <a href="/wiki/Injective_map" class="mw-redirect" title="Injective map">injective</a> topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a <a href="/wiki/Topological_embedding" class="mw-redirect" title="Topological embedding">topological embedding</a>.<a href="#cite_note-FOOTNOTEKöthe198391-2">[2]</a> A <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">topological vector space isomorphism (abbreviated <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">TVS isomorphism), also called a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">topological vector isomorphism<a href="#cite_note-FOOTNOTEGrothendieck197334–36-4">[4]</a> or an <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">isomorphism in the category of TVSs, is a bijective <a href="/wiki/Linear_map" title="Linear map">linear</a> <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>. Equivalently, it is a <a href="/wiki/Surjective_map" class="mw-redirect" title="Surjective map">surjective</a> TVS embedding<a href="#cite_note-FOOTNOTEKöthe198391-2">[2]</a> Many properties of TVSs that are studied, such as <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">local convexity</a>, <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizability</a>, <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">completeness</a>, and <a href="/wiki/Normable_space" class="mw-redirect" title="Normable space">normability</a>, are invariant under TVS isomorphisms. A necessary condition for a vector topology A collection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"> of subsets of a vector space is called additive<a href="#cite_note-FOOTNOTEWilansky201340–47-5">[5]</a> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\in {\mathcal {N}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\in {\mathcal {N}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d02dbaff32475dcf336098c8e12590904f88a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.826ex; height:2.843ex;" alt="{\displaystyle N\in {\mathcal {N}},}"> there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\in {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\in {\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3414e1e076cd4ec6ecade527ddc13c7e4353423c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.898ex; height:2.509ex;" alt="{\displaystyle U\in {\mathcal {N}}}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U+U\subseteq N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>+</mo> <mi>U</mi> <mo>⊆</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U+U\subseteq N.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e311e99725525d58bae440343abd7166404243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.214ex; height:2.343ex;" alt="{\displaystyle U+U\subseteq N.}"> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> Characterization of continuity of addition at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"><a href="#cite_note-FOOTNOTEWilansky201340–47-5">[5]</a> — If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e62d1b0c4d8d4c19a6e90bec52b49d6e36bf0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,+)}"> is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> (as all vector spaces are), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> is a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0008a48abb9837a4cb0f495dd85d0ffda22ead92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}"> is endowed with the <a href="/wiki/Product_topology" title="Product topology">product topology</a>, then the addition map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee7675af04829ff24d60d0ca986b5d909dabd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.394ex; height:2.176ex;" alt="{\displaystyle X\times X\to X}"> (defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\mapsto x+y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto x+y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5743e1f4e664dfb03d15296905ee8710bccaaee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.268ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto x+y}">) is continuous at the origin of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0008a48abb9837a4cb0f495dd85d0ffda22ead92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.8ex; height:2.176ex;" alt="{\displaystyle X\times X}"> if and only if the set of <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhoods</a> of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"> is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood." </div> All of the above conditions are consequently a necessity for a topology to form a vector topology. <div class="mw-heading mw-heading3"><h3 id="Defining_topologies_using_neighborhoods_of_the_origin">Defining topologies using neighborhoods of the origin</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=5" title="Edit section: Defining topologies using neighborhoods of the origin">edit</a>]</div> Since every vector topology is translation invariant (which means that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eaa4ab3f3f8e50c1ed796481da37454d42347c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.851ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X,}"> the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x_{0}+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x_{0}+x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4d55a7bcaac7cbc4ec3eda3c062b926d500527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.498ex; height:2.343ex;" alt="{\displaystyle x\mapsto x_{0}+x}"> is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>), to define a vector topology it suffices to define a <a href="/wiki/Neighborhood_basis" class="mw-redirect" title="Neighborhood basis">neighborhood basis</a> (or subbasis) for it at the origin. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> Theorem<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> (Neighborhood filter of the origin) — Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a real or complex vector space. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}"> is a <a href="/wiki/Empty_set" title="Empty set">non-empty</a> additive collection of <a href="/wiki/Balanced_set" title="Balanced set">balanced</a> and <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing</a> subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}"> is a <a href="/wiki/Neighborhood_base" class="mw-redirect" title="Neighborhood base">neighborhood base</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> for a vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> That is, the assumptions are that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}"> is a <a href="/wiki/Filter_base" class="mw-redirect" title="Filter base">filter base</a> that satisfies the following conditions: <ol><li>Every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0df7ead8ec7e88c3e04464929ae1213bbc1cd13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {B}}}"> is <a href="/wiki/Balanced_set" title="Balanced set">balanced</a> and <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing</a>,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}"> is additive: For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0df7ead8ec7e88c3e04464929ae1213bbc1cd13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {B}}}"> there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\in {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\in {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8538cca0bee52d35433aec1bf7b1b69adcb81fe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.167ex; height:2.176ex;" alt="{\displaystyle U\in {\mathcal {B}}}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U+U\subseteq B,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>+</mo> <mi>U</mi> <mo>⊆</mo> <mi>B</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U+U\subseteq B,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb34b9b14e2dd329f9e76e5e5f26f842f2e49dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.915ex; height:2.509ex;" alt="{\displaystyle U+U\subseteq B,}"></li></ol> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}"> satisfies the above two conditions but is not a filter base then it will form a neighborhood subbasis at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> (rather than a neighborhood basis) for a vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> </div> In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.<a href="#cite_note-FOOTNOTEWilansky201340–47-5">[5]</a> <div class="mw-heading mw-heading3"><h3 id="Defining_topologies_using_strings">Defining topologies using strings</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=6" title="Edit section: Defining topologies using strings">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be a vector space and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=1}^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd8bbf0b05c5d5b2b2dab409f8de2c1e3bcd20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.837ex; height:3.176ex;" alt="{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=1}^{\infty }}"> be a sequence of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> Each set in the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc20490b54b06d6263634b69437b4e5d6be8c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }}"> is called a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">knot of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc20490b54b06d6263634b69437b4e5d6be8c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }}"> and for every index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d0f7dadba3056fa3c06a6bee5c0b4182471152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle i,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> is called the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-th knot of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14df61a9ad3dbf1f7ccebd0320169d8e453d8f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:3.289ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }.}"> The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9e7f892894bc50c32ce1b9f9a68a15562146ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{1}}"> is called the beginning of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14df61a9ad3dbf1f7ccebd0320169d8e453d8f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:3.289ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }.}"> The sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc20490b54b06d6263634b69437b4e5d6be8c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }}"> is/is a:<a href="#cite_note-FOOTNOTEAdaschErnstKeim19785–9-7">[7]</a><a href="#cite_note-FOOTNOTESchechter1996721–751-8">[8]</a><a href="#cite_note-FOOTNOTENariciBeckenstein2011371–423-9">[9]</a> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">Summative if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>⊆</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a9e1cb6f05d4a9dfd5517ce83bf0b4c965ff67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.301ex; height:2.509ex;" alt="{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}"> for every index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffcf9ad7ad44f04fa43c5b604b4801e089981cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.449ex; height:2.176ex;" alt="{\displaystyle i.}"></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced</a> (resp. <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing</a>, closed,<a href="#cite_note-10">[note 1]</a> convex, open, <a href="/wiki/Symmetric_set" title="Symmetric set">symmetric</a>, <a href="/wiki/Barrelled_space" title="Barrelled space">barrelled</a>, <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">absolutely convex/disked</a>, etc.) if this is true of every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/212207ca3b911c3a0b9c2d77286244c917abbdb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.034ex; height:2.509ex;" alt="{\displaystyle U_{i}.}"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">String if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc20490b54b06d6263634b69437b4e5d6be8c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }}"> is summative, absorbing, and balanced.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">Topological string or a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">neighborhood string in a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc20490b54b06d6263634b69437b4e5d6be8c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }}"> is a string and each of its knots is a neighborhood of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></li></ul> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is an <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing</a> <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">disk</a> in a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then the sequence defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}:=2^{1-i}U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}:=2^{1-i}U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413ad6b49385437e1461b5551821a798e5b02b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.978ex; height:3.009ex;" alt="{\displaystyle U_{i}:=2^{1-i}U}"> forms a string beginning with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}=U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}=U.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546c07633a0e18fc2d9719bbcfb4c0e38ebaea1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.17ex; height:2.509ex;" alt="{\displaystyle U_{1}=U.}"> This is called the natural string of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"><a href="#cite_note-FOOTNOTEAdaschErnstKeim19785–9-7">[7]</a> Moreover, if a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has countable dimension then every string contains an <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">absolutely convex</a> string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued <a href="/wiki/Subadditive" class="mw-redirect" title="Subadditive">subadditive</a> functions. These functions can then be used to prove many of the basic properties of topological vector spaces. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> Theorem (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">-valued function induced by a string) — Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <msubsup> <mrow> <mo>(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7594535ebf6596000bf81f964f243bf42241cc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.837ex; height:3.176ex;" alt="{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }}"> be a collection of subsets of a vector space such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4c8f5ef6fc93255ee56e31c7c6d53d3206b7a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.39ex; height:2.509ex;" alt="{\displaystyle 0\in U_{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>⊆</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a9e1cb6f05d4a9dfd5517ce83bf0b4c965ff67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.301ex; height:2.509ex;" alt="{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\geq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f443a30b1e5e8daeed8ccfb529bc8e906dac7fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.71ex; height:2.343ex;" alt="{\displaystyle i\geq 0.}"> For all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in U_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in U_{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d015ae72a48c38867824dae78ba1669b744708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.459ex; height:2.509ex;" alt="{\displaystyle u\in U_{0},}"> let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mi>k</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>i</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>u</mi> <mo>∈</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8e27e942bbee014e1a3277deb95fbccd75285a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:79.442ex; height:3.176ex;" alt="{\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.}"> Define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbce99d737c06662e393d2d3dba7505a1e6c6eb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.462ex; height:2.843ex;" alt="{\displaystyle f:X\to [0,1]}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea78f54e69b72f398cf6077e61c50a05b532d4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=1}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\not \in U_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in U_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a25f59c54af684447945f8defb283265c550634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.812ex; height:2.676ex;" alt="{\displaystyle x\not \in U_{0}}"> and otherwise let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf4aaabe87afb886201743912835dd854f0d37f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.744ex; height:4.009ex;" alt="{\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is subadditive (meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+y)\leq f(x)+f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+y)\leq f(x)+f(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0440ac144d656f8aa4edb50a971376aad15ff3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x+y)\leq f(x)+f(y)}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}">) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigcap _{i\geq 0}U_{i};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigcap _{i\geq 0}U_{i};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5e85459584b75d08ec7c63820c30f484932734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.257ex; height:3.176ex;" alt="{\textstyle \bigcap _{i\geq 0}U_{i};}"> so in particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3f237e89ebcbd24f17125497c63b4d3749dbf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.158ex; height:2.843ex;" alt="{\displaystyle f(0)=0.}"> If all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> are <a href="/wiki/Symmetric_set" title="Symmetric set">symmetric sets</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-x)=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185fd2e78903788bc5756b067d0ac6aae1846724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.742ex; height:2.843ex;" alt="{\displaystyle f(-x)=f(x)}"> and if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> are balanced then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(sx)\leq f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(sx)\leq f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2898e266a7e937a9542ad9e025cc7a47ba35f14b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.024ex; height:2.843ex;" alt="{\displaystyle f(sx)\leq f(x)}"> for all scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |s|\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |s|\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b794f0cb3331b5fc7be8521e386157bb10e3e58f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:2.843ex;" alt="{\displaystyle |s|\leq 1}"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0deab6a01578b5b543b772df12dc0d2c593cc924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a topological vector space and if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> are neighborhoods of the origin then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is continuous, where if in addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is Hausdorff and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc20490b54b06d6263634b69437b4e5d6be8c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }}"> forms a basis of balanced neighborhoods of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y):=f(x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y):=f(x-y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00809bc74240044371f51fff671c20287370acdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.703ex; height:2.843ex;" alt="{\displaystyle d(x,y):=f(x-y)}"> is a metric defining the vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> </div> A proof of the above theorem is given in the article on <a href="/wiki/Metrizable_topological_vector_space#Additive_sequences" title="Metrizable topological vector space">metrizable topological vector spaces</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i\in \mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1756d8e22b11725e66e8bdaba7a69ab5457fe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.019ex; height:3.009ex;" alt="{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i\in \mathbb {N} }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\bullet }=\left(V_{i}\right)_{i\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\bullet }=\left(V_{i}\right)_{i\in \mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/947f739b4a5ca0cfb701d26a181f0b322e5b16a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.555ex; height:3.009ex;" alt="{\displaystyle V_{\bullet }=\left(V_{i}\right)_{i\in \mathbb {N} }}"> are two collections of subsets of a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> is a scalar, then by definition:<a href="#cite_note-FOOTNOTEAdaschErnstKeim19785–9-7">[7]</a> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa954d412ca56c0e1d4ebe7ae5f9fb17ddc67bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{\bullet }}"> contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc20490b54b06d6263634b69437b4e5d6be8c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{\bullet }}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ U_{\bullet }\subseteq V_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>⊆</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ U_{\bullet }\subseteq V_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dbed5593fad2f40610691111daf9f9d40268e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.323ex; margin-bottom: -0.348ex; width:8.73ex; height:2.509ex;" alt="{\displaystyle \ U_{\bullet }\subseteq V_{\bullet }}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}\subseteq V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}\subseteq V_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97e7168c76a75cc453ad24cdddc1e3ae6b22392d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.641ex; height:2.509ex;" alt="{\displaystyle U_{i}\subseteq V_{i}}"> for every index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffcf9ad7ad44f04fa43c5b604b4801e089981cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.449ex; height:2.176ex;" alt="{\displaystyle i.}"></li> <li>Set of knots: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \operatorname {Knots} U_{\bullet }:=\left\{U_{i}:i\in \mathbb {N} \right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Knots</mi> <mo>⁡</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \operatorname {Knots} U_{\bullet }:=\left\{U_{i}:i\in \mathbb {N} \right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/452eae40cbed40d8a074bbd8acd922bb359d49ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.83ex; height:2.843ex;" alt="{\displaystyle \ \operatorname {Knots} U_{\bullet }:=\left\{U_{i}:i\in \mathbb {N} \right\}.}"></li> <li>Kernel: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ \ker U_{\bullet }:=\bigcap _{i\in \mathbb {N} }U_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> </mtext> <mi>ker</mi> <mo>⁡</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:=</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \ \ker U_{\bullet }:=\bigcap _{i\in \mathbb {N} }U_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bd98d3400b701f8a389f26e26842dabd07d43c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.353ex; height:3.009ex;" alt="{\textstyle \ \ker U_{\bullet }:=\bigcap _{i\in \mathbb {N} }U_{i}.}"></li> <li>Scalar multiple: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ sU_{\bullet }:=\left(sU_{i}\right)_{i\in \mathbb {N} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>s</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:=</mo> <msub> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ sU_{\bullet }:=\left(sU_{i}\right)_{i\in \mathbb {N} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93f2f8cb0969059498598839c2e2b9afef90a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.074ex; height:3.009ex;" alt="{\displaystyle \ sU_{\bullet }:=\left(sU_{i}\right)_{i\in \mathbb {N} }.}"></li> <li>Sum: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ U_{\bullet }+V_{\bullet }:=\left(U_{i}+V_{i}\right)_{i\in \mathbb {N} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:=</mo> <msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ U_{\bullet }+V_{\bullet }:=\left(U_{i}+V_{i}\right)_{i\in \mathbb {N} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78beff5451ebccb511fc5be0d0859ae6c69521e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.139ex; height:3.009ex;" alt="{\displaystyle \ U_{\bullet }+V_{\bullet }:=\left(U_{i}+V_{i}\right)_{i\in \mathbb {N} }.}"></li> <li>Intersection: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ U_{\bullet }\cap V_{\bullet }:=\left(U_{i}\cap V_{i}\right)_{i\in \mathbb {N} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>∩</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>:=</mo> <msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ U_{\bullet }\cap V_{\bullet }:=\left(U_{i}\cap V_{i}\right)_{i\in \mathbb {N} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d734d956f5b6af7fb0da09297648b98c16cdd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.623ex; height:3.009ex;" alt="{\displaystyle \ U_{\bullet }\cap V_{\bullet }:=\left(U_{i}\cap V_{i}\right)_{i\in \mathbb {N} }.}"></li></ul> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> is a collection sequences of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> is said to be directed (downwards) under inclusion or simply directed downward if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> is not empty and for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\bullet },V_{\bullet }\in \mathbb {S} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\bullet },V_{\bullet }\in \mathbb {S} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d45d717de97e210bf6c94236f5afb96acb28d25c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.865ex; height:2.509ex;" alt="{\displaystyle U_{\bullet },V_{\bullet }\in \mathbb {S} ,}"> there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\bullet }\in \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\bullet }\in \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77cc14d36368d44c01191a47eba65877520bdb2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:7.381ex; height:2.509ex;" alt="{\displaystyle W_{\bullet }\in \mathbb {S} }"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\bullet }\subseteq U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>⊆</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\bullet }\subseteq U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b052255cfaa40c54fc17e1db5259d0a79f7ab904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.323ex; margin-bottom: -0.348ex; width:8.988ex; height:2.509ex;" alt="{\displaystyle W_{\bullet }\subseteq U_{\bullet }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\bullet }\subseteq V_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>⊆</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\bullet }\subseteq V_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136da9ef63ec13a84fac00200eb594440c1c0ed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.323ex; margin-bottom: -0.348ex; width:8.756ex; height:2.509ex;" alt="{\displaystyle W_{\bullet }\subseteq V_{\bullet }}"> (said differently, if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> is a <a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">prefilter</a> with respect to the containment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\subseteq \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>⊆</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\subseteq \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591976ad7ed25b287998b2c438d5391be58c5c18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,\subseteq \,}"> defined above). Notation: Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Knots} \mathbb {S} :=\bigcup _{U_{\bullet }\in \mathbb {S} }\operatorname {Knots} U_{\bullet }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Knots</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mo>:=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mrow> </munder> <mi>Knots</mi> <mo>⁡</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {Knots} \mathbb {S} :=\bigcup _{U_{\bullet }\in \mathbb {S} }\operatorname {Knots} U_{\bullet }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b41a9ae0d4474889dfbc88ba2464076e332f2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-bottom: -0.333ex; width:27.141ex; height:3.176ex;" alt="{\textstyle \operatorname {Knots} \mathbb {S} :=\bigcup _{U_{\bullet }\in \mathbb {S} }\operatorname {Knots} U_{\bullet }}"> be the set of all knots of all strings in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/522b002a7daa6a8961cf43aa78e471df51ae3e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} .}"> Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> Theorem<a href="#cite_note-FOOTNOTEAdaschErnstKeim19785–9-7">[7]</a> (Topology induced by strings) — If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"> is a topological vector space then there exists a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"><a href="#cite_note-11">[proof 1]</a> of neighborhood strings in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that is directed downward and such that the set of all knots of all strings in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> is a <a href="/wiki/Neighborhood_basis" class="mw-redirect" title="Neighborhood basis">neighborhood basis</a> at the origin for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b42cadba3d621e09a00a4e133d83ad19f6d548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle (X,\tau ).}"> Such a collection of strings is said to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> fundamental. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a vector space and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> is a collection of strings in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that is directed downward, then the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Knots} \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Knots</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Knots} \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2203077da9ddba238f787c58a5d80af1052432d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.764ex; height:2.176ex;" alt="{\displaystyle \operatorname {Knots} \mathbb {S} }"> of all knots of all strings in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> forms a <a href="/wiki/Neighborhood_basis" class="mw-redirect" title="Neighborhood basis">neighborhood basis</a> at the origin for a vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> In this case, this topology is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\mathbb {S} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\mathbb {S} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70e9eac5d0327b933b6981fcdfde5a7d9fde3f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.162ex; height:2.009ex;" alt="{\displaystyle \tau _{\mathbb {S} }}"> and it is called the topology generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/522b002a7daa6a8961cf43aa78e471df51ae3e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} .}"> </div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> is the set of all topological strings in a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\mathbb {S} }=\tau .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\mathbb {S} }=\tau .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2bedae9243ec4f003a0821f4fb19c36ec4b74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.11ex; height:2.009ex;" alt="{\displaystyle \tau _{\mathbb {S} }=\tau .}"><a href="#cite_note-FOOTNOTEAdaschErnstKeim19785–9-7">[7]</a> A Hausdorff TVS is <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizable</a> if and only if its topology can be induced by a single topological string.<a href="#cite_note-FOOTNOTEAdaschErnstKeim197810–15-12">[10]</a> <div class="mw-heading mw-heading2"><h2 id="Topological_structure">Topological structure</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=7" title="Edit section: Topological structure">edit</a>]</div> A vector space is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">). Hence, every topological vector space is an abelian <a href="/wiki/Topological_group" title="Topological group">topological group</a>. Every TVS is <a href="/wiki/Completely_regular" class="mw-redirect" title="Completely regular">completely regular</a> but a TVS need not be <a href="/wiki/Normal_space" title="Normal space">normal</a>.<a href="#cite_note-FOOTNOTEWilansky201353-13">[11]</a> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be a topological vector space. Given a <a href="/wiki/Subspace_topology" title="Subspace topology">subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\subseteq X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70195cdd6acbfffcc88f32f5e827339575f85698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.168ex; height:2.509ex;" alt="{\displaystyle M\subseteq X,}"> the quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aaa4313768efcc32d3d5a1de282d7bbe08baec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.585ex; height:2.843ex;" alt="{\displaystyle X/M}"> with the usual <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient topology</a> is a Hausdorff topological vector space if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is closed.<a href="#cite_note-14">[note 2]</a> This permits the following construction: given a topological vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (that is probably not Hausdorff), form the quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aaa4313768efcc32d3d5a1de282d7bbe08baec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.585ex; height:2.843ex;" alt="{\displaystyle X/M}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is the closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5092383232dea9773e6f99baaa92bb31a25f1b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \{0\}.}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aaa4313768efcc32d3d5a1de282d7bbe08baec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.585ex; height:2.843ex;" alt="{\displaystyle X/M}"> is then a Hausdorff topological vector space that can be studied instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> <div class="mw-heading mw-heading3"><h3 id="Invariance_of_vector_topologies">Invariance of vector topologies</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=8" title="Edit section: Invariance of vector topologies">edit</a>]</div> One of the most used properties of vector topologies is that every vector topology is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">translation invariant: <dl><dd>for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eaa4ab3f3f8e50c1ed796481da37454d42347c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.851ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X,}"> the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x_{0}+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x_{0}+x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4d55a7bcaac7cbc4ec3eda3c062b926d500527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.498ex; height:2.343ex;" alt="{\displaystyle x\mapsto x_{0}+x}"> is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>, but if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c873d33292c8b8b4f57edbade8876f4894a5a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x_{0}\neq 0}"> then it is not linear and so not a TVS-isomorphism.</dd></dl> Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b696e0b457eecf592be8e2fda417810807613c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.351ex; height:2.676ex;" alt="{\displaystyle s\neq 0}"> then the linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto sx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>s</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto sx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b19ee96f34f931bce31787dd2890dc2d9c671c76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.364ex; height:1.843ex;" alt="{\displaystyle x\mapsto sx}"> is a homeomorphism. Using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999944118796b0e4485e997249775b0d9925772f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.16ex; height:2.343ex;" alt="{\displaystyle s=-1}"> produces the negation map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle X\to X}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto -x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto -x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99f8556c8a904b2285cb3d1bd0fc55c6fef1c264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.728ex; height:2.343ex;" alt="{\displaystyle x\mapsto -x,}"> which is consequently a linear homeomorphism and thus a TVS-isomorphism. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"> and any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcee64dbb80f9474630b849a15567dc2acd95608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle S\subseteq X,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(x+S)=x+\operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(x+S)=x+\operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc41a023028d9da9978141d4bd351f842af21f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.257ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(x+S)=x+\operatorname {cl} _{X}S}"><a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> and moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/415e70bacad715c20d96098344bf05eb5a064fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.502ex; height:2.176ex;" alt="{\displaystyle 0\in S}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796e5d27fc0baac03f543ac8b92ff0299a6f7842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.669ex; height:2.343ex;" alt="{\displaystyle x+S}"> is a <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> (resp. open neighborhood, closed neighborhood) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> if and only if the same is true of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> at the origin. <div class="mw-heading mw-heading3"><h3 id="Local_notions">Local notions</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=9" title="Edit section: Local notions">edit</a>]</div> A subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> of a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is said to be <ul><li><a href="/wiki/Absorbing_set" title="Absorbing set">absorbing</a> (in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">): if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}"> there exists a real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>0}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cx\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cx\in E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61eabbfd2e9437365a498d1f417312140233b226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.953ex; height:2.176ex;" alt="{\displaystyle cx\in E}"> for any scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |c|\leq r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c|\leq r.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cba3d1ce0c07d3a48bef204fa8344a07d6c6f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.094ex; height:2.843ex;" alt="{\displaystyle |c|\leq r.}"><a href="#cite_note-FOOTNOTERudin19916_§1.4-15">[12]</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">balanced</a> or circled: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tE\subseteq E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>E</mi> <mo>⊆</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tE\subseteq E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19192b264b7bbb6c3324580f47b8e4b13245941e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.489ex; height:2.343ex;" alt="{\displaystyle tE\subseteq E}"> for every scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|\leq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |t|\leq 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167c3a7153e332880808c0fe0ae76ff29975027d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.041ex; height:2.843ex;" alt="{\displaystyle |t|\leq 1.}"><a href="#cite_note-FOOTNOTERudin19916_§1.4-15">[12]</a></li> <li><a href="/wiki/Convex_set" title="Convex set">convex</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tE+(1-t)E\subseteq E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>E</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo>⊆</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tE+(1-t)E\subseteq E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef7ff2a6644e4fddd18bebb216935e7437a5a11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.757ex; height:2.843ex;" alt="{\displaystyle tE+(1-t)E\subseteq E}"> for every real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq t\leq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq t\leq 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de2f7f2acd36689fdbad7f715c93b7c69006fbcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.008ex; height:2.343ex;" alt="{\displaystyle 0\leq t\leq 1.}"><a href="#cite_note-FOOTNOTERudin19916_§1.4-15">[12]</a></li> <li>a <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">disk</a> or <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">absolutely convex</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is convex and balanced.</li> <li><a href="/wiki/Symmetric_set" title="Symmetric set">symmetric</a>: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -E\subseteq E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>E</mi> <mo>⊆</mo> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E\subseteq E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c93cee4e1597c853ff4112d0f68ddcf6e420663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.105ex; height:2.509ex;" alt="{\displaystyle -E\subseteq E,}"> or equivalently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -E=E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>E</mi> <mo>=</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E=E.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d32975eba962f5c4c28c4e4f5c88c3be16e60a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.105ex; height:2.343ex;" alt="{\displaystyle -E=E.}"></li></ul> Every neighborhood of the origin is an <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing set</a> and contains an open <a href="/wiki/Balanced_set" title="Balanced set">balanced</a> neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"><a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> so every topological vector space has a local base of absorbing and <a href="/wiki/Balanced_set" title="Balanced set">balanced sets</a>. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b706e1a4adcd327c3b9ed05ce689efa065b39019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 0;}"> if the space is <a href="/wiki/Locally_convex" class="mw-redirect" title="Locally convex">locally convex</a> then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin. Bounded subsets A subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> of a topological vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">bounded</a><a href="#cite_note-FOOTNOTERudin19918-16">[13]</a> if for every neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> of the origin there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq tV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mi>t</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq tV}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc962b70bc6363a30ed5c0da02f67c5deb59fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.501ex; height:2.343ex;" alt="{\displaystyle E\subseteq tV}">. The definition of boundedness can be weakened a bit; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set.<a href="#cite_note-FOOTNOTENariciBeckenstein2011155–176-17">[14]</a> Also, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is bounded if and only if for every balanced neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> of the origin, there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq tV.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mi>t</mi> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq tV.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65ab433004241352550afe333ebb728d815a79b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.148ex; height:2.343ex;" alt="{\displaystyle E\subseteq tV.}"> Moreover, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is locally convex, the boundedness can be characterized by <a href="/wiki/Seminorm" title="Seminorm">seminorms</a>: the subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is bounded if and only if every continuous seminorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is bounded on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2566d01f104ef084ea424b8b35c2534f7f902b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}"><a href="#cite_note-FOOTNOTERudin199127-28_Theorem_1.37-18">[15]</a> Every <a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a> set is bounded.<a href="#cite_note-FOOTNOTENariciBeckenstein2011155–176-17">[14]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a vector subspace of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> then a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is bounded in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> if and only if it is bounded in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"><a href="#cite_note-FOOTNOTENariciBeckenstein2011155–176-17">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Metrizability">Metrizability</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=10" title="Edit section: Metrizability">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <a href="/wiki/Birkhoff%E2%80%93Kakutani_theorem" class="mw-redirect" title="Birkhoff–Kakutani theorem">Birkhoff–Kakutani theorem</a> — If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"> is a topological vector space then the following four conditions are equivalent:<a href="#cite_note-FOOTNOTEKöthe1983section_15.11-19">[16]</a><a href="#cite_note-20">[note 3]</a> <ol><li>The origin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and there is a <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> <a href="/wiki/Neighborhood_basis" class="mw-redirect" title="Neighborhood basis">basis of neighborhoods</a> at the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"> is <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable</a> (as a topological space).</li> <li>There is a <a href="/wiki/Translation-invariant_metric" class="mw-redirect" title="Translation-invariant metric">translation-invariant metric</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that induces on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> the topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26d6cc28c28ff4ff88402f47f2a99e583e9e045f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.849ex; height:2.009ex;" alt="{\displaystyle \tau ,}"> which is the given topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"> is a <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizable topological vector space</a>.<a href="#cite_note-21">[note 4]</a></li></ol> By the Birkhoff–Kakutani theorem, it follows that there is an <a href="/wiki/Equivalence_of_metrics" title="Equivalence of metrics">equivalent metric</a> that is translation-invariant. </div> A TVS is <a href="/wiki/Metrizable_TVS" class="mw-redirect" title="Metrizable TVS">pseudometrizable</a> if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an <a href="/wiki/Metrizable_TVS" class="mw-redirect" title="Metrizable TVS">F-seminorm</a>. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable. More strongly: a topological vector space is said to be <a href="/wiki/Normable" class="mw-redirect" title="Normable">normable</a> if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin.<a href="#cite_note-springer-22">[17]</a> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> be a non-<a href="/wiki/Discrete_space" title="Discrete space">discrete</a> <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> topological field, for example the real or complex numbers. A <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> topological vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> is locally compact if and only if it is <a href="/wiki/Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a>, that is, isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8ae3da7ba82a494455bdb6b113004b453be41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {K} ^{n}}"> for some natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"><a href="#cite_note-FOOTNOTERudin199117_Theorem_1.22-23">[18]</a> <div class="mw-heading mw-heading3"><h3 id="Completeness_and_uniform_structure">Completeness and uniform structure</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=11" title="Edit section: Completeness and uniform structure">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete topological vector space</a></div> The <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">canonical uniformity</a><a href="#cite_note-FOOTNOTESchaeferWolff199912–19-24">[19]</a> on a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"> is the unique translation-invariant <a href="/wiki/Uniform_space" title="Uniform space">uniformity</a> that induces the topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into <a href="/wiki/Uniform_space" title="Uniform space">uniform spaces</a>. This allows one to talk[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] about related notions such as <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">completeness</a>, <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a>, Cauchy nets, and <a href="/wiki/Uniform_continuity" title="Uniform continuity">uniform continuity</a>, etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is <a href="/wiki/Tychonoff_space" title="Tychonoff space">Tychonoff</a>.<a href="#cite_note-FOOTNOTESchaeferWolff199916-25">[20]</a> A subset of a TVS is <a href="/wiki/Compact_space" title="Compact space">compact</a> if and only if it is complete and <a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a> (for Hausdorff TVSs, a set being totally bounded is equivalent to it being <a href="/wiki/Totally_bounded_space#In_topological_groups" title="Totally bounded space">precompact</a>). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are <a href="/wiki/Relatively_compact" class="mw-redirect" title="Relatively compact">relatively compact</a>). With respect to this uniformity, a <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">net</a> (or <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f06cbd9ae8f0977830b0571545f637115665c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.146ex; height:3.009ex;" alt="{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}"> is Cauchy if and only if for every neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95547343453ea34a314dd174f8458012f5a39ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 0,}"> there exists some index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}-x_{j}\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}-x_{j}\in V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d041746ac0e39de2be17ab96421eed5cbb02d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.837ex; height:2.843ex;" alt="{\displaystyle x_{i}-x_{j}\in V}"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\geq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≥</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\geq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/515060afe2c7efb26b4257f4f89cd8526d86a0f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.296ex; height:2.343ex;" alt="{\displaystyle i\geq n}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\geq n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>≥</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\geq n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a03df68092f484e6987def399339436d1e522403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:6.125ex; height:2.509ex;" alt="{\displaystyle j\geq n.}"> Every Cauchy sequence is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called <a href="/wiki/Sequentially_complete" title="Sequentially complete">sequentially complete</a>; in general, it may not be complete (in the sense that all Cauchy filters converge). The vector space operation of addition is uniformly continuous and an <a href="/w/index.php?title=Open_and_closed_map&action=edit&redlink=1" class="new" title="Open and closed map (page does not exist)">open map</a>. Scalar multiplication is <a href="/wiki/Cauchy_continuous" class="mw-redirect" title="Cauchy continuous">Cauchy continuous</a> but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a <a href="/wiki/Dense_set" title="Dense set">dense</a> <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of a <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete topological vector space</a>. <ul><li>Every TVS has a <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">completion</a> and every Hausdorff TVS has a Hausdorff completion.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions.</li> <li>A compact subset of a TVS (not necessarily Hausdorff) is complete.<a href="#cite_note-FOOTNOTENariciBeckenstein2011115–154-26">[21]</a> A complete subset of a Hausdorff TVS is closed.<a href="#cite_note-FOOTNOTENariciBeckenstein2011115–154-26">[21]</a></li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is a complete subset of a TVS then any subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> that is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is complete.<a href="#cite_note-FOOTNOTENariciBeckenstein2011115–154-26">[21]</a></li> <li>A Cauchy sequence in a Hausdorff TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is not necessarily <a href="/wiki/Relatively_compact" class="mw-redirect" title="Relatively compact">relatively compact</a> (that is, its closure in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is not necessarily compact).</li> <li>If a Cauchy filter in a TVS has an <a href="/wiki/Filters_in_topology" title="Filters in topology">accumulation point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> then it converges to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}"></li> <li>If a series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{\infty }x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{\infty }x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83e196df2a8b33e6fb4bda0c77da38ed193ba12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.87ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{\infty }x_{i}}"> converges<a href="#cite_note-27">[note 5]</a> in a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\bullet }\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\bullet }\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/937def0f95a8d85567d6557bb38ab5d59190d679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:7.16ex; height:2.509ex;" alt="{\displaystyle x_{\bullet }\to 0}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"><a href="#cite_note-FOOTNOTESwartz199227–29-28">[22]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=12" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Finest_and_coarsest_vector_topology">Finest and coarsest vector topology</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=13" title="Edit section: Finest and coarsest vector topology">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be a real or complex vector space. Trivial topology The <a href="/wiki/Trivial_topology" title="Trivial topology">trivial topology</a> or indiscrete topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X,\varnothing \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo>,</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X,\varnothing \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26c6c5c4dbc65daf37dedc4f9a4646801abdaf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.147ex; height:2.843ex;" alt="{\displaystyle \{X,\varnothing \}}"> is always a TVS topology on any vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a>) <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete</a> <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">pseudometrizable</a> <a href="/wiki/Seminormed_space" class="mw-redirect" title="Seminormed space">seminormable</a> <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex</a> topological vector space. It is <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim X=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim X=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe73254a77f290f0cd85f258ea803ad063403ebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.15ex; height:2.176ex;" alt="{\displaystyle \dim X=0.}"> Finest vector topology There exists a TVS topology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a776a241902fe9a448161d4e738267f54a1799f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.152ex; height:2.343ex;" alt="{\displaystyle \tau _{f}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">finest vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> that is finer than every other TVS-topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (that is, any TVS-topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is necessarily a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a776a241902fe9a448161d4e738267f54a1799f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.152ex; height:2.343ex;" alt="{\displaystyle \tau _{f}}">).<a href="#cite_note-29">[23]</a><a href="#cite_note-FOOTNOTENariciBeckenstein2011111-30">[24]</a> Every linear map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X,\tau _{f}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(X,\tau _{f}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1b2d6082219bd64c2c1a9f5eeb679405b5978c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.976ex; height:3.009ex;" alt="{\displaystyle \left(X,\tau _{f}\right)}"> into another TVS is necessarily continuous. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has an uncountable <a href="/wiki/Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel basis</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{f}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a776a241902fe9a448161d4e738267f54a1799f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.152ex; height:2.343ex;" alt="{\displaystyle \tau _{f}}"> is not <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex</a> and not <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizable</a>.<a href="#cite_note-FOOTNOTENariciBeckenstein2011111-30">[24]</a> <div class="mw-heading mw-heading3"><h3 id="Cartesian_products">Cartesian products</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=14" title="Edit section: Cartesian products">edit</a>]</div> A <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of a family of topological vector spaces, when endowed with the <a href="/wiki/Product_topology" title="Product topology">product topology</a>, is a topological vector space. Consider for instance the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> of all functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a10a3ad05781f5cf9c2d875a02227e21a8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> carries its usual <a href="/wiki/Euclidean_topology" title="Euclidean topology">Euclidean topology</a>. This set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\mathbb {R} },,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msup> <mo>,</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\mathbb {R} },,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca91d97129d5c68b538b82fd1991859f0e0f0d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.778ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{\mathbb {R} },,}"> which carries the natural <a href="/wiki/Product_topology" title="Product topology">product topology</a>. With this product topology, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\mathbb {R} ^{\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:=\mathbb {R} ^{\mathbb {R} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca196e8b7d75337a149ed30e0939e19e47993698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.822ex; height:2.676ex;" alt="{\displaystyle X:=\mathbb {R} ^{\mathbb {R} }}"> becomes a topological vector space whose topology is called the topology of <a href="/wiki/Pointwise_convergence" title="Pointwise convergence">pointwise convergence</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}"> The reason for this name is the following: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f_{n}\right)_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(f_{n}\right)_{n=1}^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996653858127ce1cabd8e93c889288c6efc6a90a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.486ex; height:3.176ex;" alt="{\displaystyle \left(f_{n}\right)_{n=1}^{\infty }}"> is a <a href="/wiki/Sequence" title="Sequence">sequence</a> (or more generally, a <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">net</a>) of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/888dd9afe888206b1b0a5cd2df0b7597bf5ada1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:2.509ex;" alt="{\displaystyle f\in X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2702450f0458a5e01a698e248af552a7fab2b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}"> <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converges</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> if and only if for every real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86f950ad1f8135017742e12a8da5e2367bfdc1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.497ex; height:2.843ex;" alt="{\displaystyle f_{n}(x)}"> converges to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}"> This TVS is <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete</a>, <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, and <a href="/wiki/Locally_convex" class="mw-redirect" title="Locally convex">locally convex</a> but not <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizable</a> and consequently not <a href="/wiki/Normable" class="mw-redirect" title="Normable">normable</a>; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} f:=\{rf:r\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mi>f</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>r</mi> <mi>f</mi> <mo>:</mo> <mi>r</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} f:=\{rf:r\in \mathbb {R} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7948c817d8504c679944feac234ef902f93937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.859ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} f:=\{rf:r\in \mathbb {R} \}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88725631c1c1c441d1b0db0aed0e22246f162b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.54ex; height:2.676ex;" alt="{\displaystyle f\neq 0}">). <div class="mw-heading mw-heading3"><h3 id="Finite-dimensional_spaces">Finite-dimensional spaces</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=15" title="Edit section: Finite-dimensional spaces">edit</a>]</div> By <a href="/wiki/F._Riesz%27s_theorem" title="F. Riesz's theorem">F. Riesz's theorem</a>, a Hausdorff topological vector space is finite-dimensional if and only if it is <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a>, which happens if and only if it has a compact <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> of the origin. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> and endow <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> with its usual Hausdorff normed <a href="/wiki/Euclidean_topology" title="Euclidean topology">Euclidean topology</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be a vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> of finite dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:=\dim X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>:=</mo> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n:=\dim X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd5300918a57cbaf3bf18f684d446434dddc79c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.382ex; height:2.176ex;" alt="{\displaystyle n:=\dim X}"> and so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is vector space isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8ae3da7ba82a494455bdb6b113004b453be41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {K} ^{n}}"> (explicitly, this means that there exists a <a href="/wiki/Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">linear isomorphism</a> between the vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8ae3da7ba82a494455bdb6b113004b453be41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {K} ^{n}}">). This finite-dimensional vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> always has a unique <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> vector topology, which makes it TVS-isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1b0e9deab96680361150c03545251d39f90634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.673ex; height:2.676ex;" alt="{\displaystyle \mathbb {K} ^{n},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8ae3da7ba82a494455bdb6b113004b453be41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.027ex; height:2.343ex;" alt="{\displaystyle \mathbb {K} ^{n}}"> is endowed with the usual Euclidean topology (which is the same as the <a href="/wiki/Product_topology" title="Product topology">product topology</a>). This Hausdorff vector topology is also the (unique) <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">finest</a> vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has a unique vector topology if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim X=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim X=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe73254a77f290f0cd85f258ea803ad063403ebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.15ex; height:2.176ex;" alt="{\displaystyle \dim X=0.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim X\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim X\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53313d217d73cb0c003bbb58ccaf37fa06f8f6f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.503ex; height:2.676ex;" alt="{\displaystyle \dim X\neq 0}"> then although <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> does not have a unique vector topology, it does have a unique Hausdorff vector topology. <ul><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim X=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim X=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f791015b08019c26cd66f6a9d28e447637cb038c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.503ex; height:2.176ex;" alt="{\displaystyle \dim X=0}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb567271e86f10b004e7afbb7a348acc7691a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.566ex; height:2.843ex;" alt="{\displaystyle X=\{0\}}"> has exactly one vector topology: the <a href="/wiki/Trivial_topology" title="Trivial topology">trivial topology</a>, which in this case (and only in this case) is Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}"></li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim X=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim X=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df88c0fd7f895747460e3706093fd9203a754ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.503ex; height:2.176ex;" alt="{\displaystyle \dim X=1}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has two vector topologies: the usual <a href="/wiki/Euclidean_topology" title="Euclidean topology">Euclidean topology</a> and the (non-Hausdorff) trivial topology. <ul><li>Since the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> is itself a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">-dimensional topological vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing set</a> and has consequences that reverberate throughout <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.</li></ul></li></ul> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style="">Proof outline The proof of this dichotomy (i.e. that a vector topology is either trivial or isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }">) is straightforward so only an outline with the important observations is given. As usual, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> is assumed have the (normed) Euclidean topology. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}:=\{a\in \mathbb {K} :|a|<r\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}:=\{a\in \mathbb {K} :|a|<r\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c94a830513b624a943d0717f75a3815965135a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.294ex; height:2.843ex;" alt="{\displaystyle B_{r}:=\{a\in \mathbb {K} :|a|<r\}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee752d0040bb7b4c93ef4df2a7956ec088a32c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.956ex; height:2.176ex;" alt="{\displaystyle r>0.}"> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">-dimensional vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a85c73bd0f34d264a8b2a361a14e66ab0231834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} .}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3991ba161c3d9db0e47b1101a46dd0c2e70f51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.671ex; height:2.343ex;" alt="{\displaystyle B\subseteq \mathbb {K} }"> is a ball centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\cdot S=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⋅</mo> <mi>S</mi> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\cdot S=X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4441d1bc19ca4a04d9bbfbf39ea20de354895e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.021ex; height:2.176ex;" alt="{\displaystyle B\cdot S=X}"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> contains an "unbounded sequence", by which it is meant a sequence of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{i}x\right)_{i=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a_{i}x\right)_{i=1}^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b632c49d3042fba2774297a9ece9db6ae1ec1f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.069ex; height:3.176ex;" alt="{\displaystyle \left(a_{i}x\right)_{i=1}^{\infty }}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≠</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\neq x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/131de40518d9969e9e9391fbc6ede601d44fd28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.411ex; height:2.676ex;" alt="{\displaystyle 0\neq x\in X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{i}\right)_{i=1}^{\infty }\subseteq \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a_{i}\right)_{i=1}^{\infty }\subseteq \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9e315e24ca6696233e4a77320e93313f997831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.646ex; height:3.176ex;" alt="{\displaystyle \left(a_{i}\right)_{i=1}^{\infty }\subseteq \mathbb {K} }"> is unbounded in normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"> (in the usual sense). Any vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> will be translation invariant and invariant under non-zero scalar multiplication, and for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≠</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\neq x\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce1978d347ee241fdb0f3ae1d49d4669b7f2563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.058ex; height:2.676ex;" alt="{\displaystyle 0\neq x\in X,}"> the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{x}:\mathbb {K} \to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{x}:\mathbb {K} \to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/becf78fc5221ed06de45e7bc6172d62b3557a867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.766ex; height:2.509ex;" alt="{\displaystyle M_{x}:\mathbb {K} \to X}"> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{x}(a):=ax}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>a</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{x}(a):=ax}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccbd31bb8f655ae46e0a22fd0651dc80a8ae66c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.77ex; height:2.843ex;" alt="{\displaystyle M_{x}(a):=ax}"> is a continuous linear bijection. Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {K} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbb {K} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08996d626a188c132c1a59abfbe1dc8cf3c37a90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.216ex; height:2.176ex;" alt="{\displaystyle X=\mathbb {K} x}"> for any such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> every subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Fx=M_{x}(F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Fx=M_{x}(F)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffdbc965ed71abf311d10e217019f7b3b92edee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.146ex; height:2.843ex;" alt="{\displaystyle Fx=M_{x}(F)}"> for some unique subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subseteq \mathbb {K} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subseteq \mathbb {K} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8db2b643c447c1ebaf9b59293403909711668a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.294ex; height:2.343ex;" alt="{\displaystyle F\subseteq \mathbb {K} .}"> And if this vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has a neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"> of the origin that is not equal to all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> then the continuity of scalar multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} \times X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} \times X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b538d19d550277121edb66ac309032becd50b740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.223ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} \times X\to X}"> at the origin guarantees the existence of an open ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}\subseteq \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}\subseteq \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76506cd5afa7c419e43655f0ce3ed2cd928c3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.644ex; height:2.509ex;" alt="{\displaystyle B_{r}\subseteq \mathbb {K} }"> centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> and an open neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}\cdot S\subseteq W\neq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⋅</mo> <mi>S</mi> <mo>⊆</mo> <mi>W</mi> <mo>≠</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}\cdot S\subseteq W\neq X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f419d66d8b038bd9e33f03904c53cea55d9e830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.175ex; height:2.676ex;" alt="{\displaystyle B_{r}\cdot S\subseteq W\neq X,}"> which implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> does not contain any "unbounded sequence". This implies that for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≠</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\neq x\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce1978d347ee241fdb0f3ae1d49d4669b7f2563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.058ex; height:2.676ex;" alt="{\displaystyle 0\neq x\in X,}"> there exists some positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq B_{n}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq B_{n}x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641dcdb9df1c2b8c52039b0a5c72f698a1d1ee3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.557ex; height:2.509ex;" alt="{\displaystyle S\subseteq B_{n}x.}"> From this, it can be deduced that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> does not carry the trivial topology and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq x\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≠</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\neq x\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce1978d347ee241fdb0f3ae1d49d4669b7f2563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.058ex; height:2.676ex;" alt="{\displaystyle 0\neq x\in X,}"> then for any ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3991ba161c3d9db0e47b1101a46dd0c2e70f51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.671ex; height:2.343ex;" alt="{\displaystyle B\subseteq \mathbb {K} }"> center at 0 in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07453cff88535372e3ebb320517acf77be68529d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {K} ,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{x}(B)=Bx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>B</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{x}(B)=Bx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44883d1badc46c2f409f8f5f58047de7c2cc59d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.192ex; height:2.843ex;" alt="{\displaystyle M_{x}(B)=Bx}"> contains an open neighborhood of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> which then proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffb6beda78e4e8a06810b7923be306ef5376c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.427ex; height:2.509ex;" alt="{\displaystyle M_{x}}"> is a linear <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>. <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8733090f2d787d03101c3e16dc3f6404f0e7dd4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }"> </div> <ul><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim X=n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>=</mo> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim X=n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca033ec4414426959d658304f1dce2808c76264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.996ex; height:2.343ex;" alt="{\displaystyle \dim X=n\geq 2}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has infinitely many distinct vector topologies: <ul><li>Some of these topologies are now described: Every linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> which is vector space isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1b0e9deab96680361150c03545251d39f90634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.673ex; height:2.676ex;" alt="{\displaystyle \mathbb {K} ^{n},}"> induces a <a href="/wiki/Seminorm" title="Seminorm">seminorm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|:X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|:X\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f491a934ac17fa8be614263eb4d2b60115e337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.782ex; height:2.843ex;" alt="{\displaystyle |f|:X\to \mathbb {R} }"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|(x)=|f(x)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|(x)=|f(x)|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8fd7b147803d78716da7d712d39bd89373f1b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.521ex; height:2.843ex;" alt="{\displaystyle |f|(x)=|f(x)|}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker f=\ker |f|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mi>f</mi> <mo>=</mo> <mi>ker</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker f=\ker |f|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995d75b20c77c8cae02ef2e36a20850dd063d710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.713ex; height:2.843ex;" alt="{\displaystyle \ker f=\ker |f|.}"> Every seminorm induces a (<a href="/wiki/Metrizable_TVS" class="mw-redirect" title="Metrizable TVS">pseudometrizable</a> <a href="/wiki/Locally_convex" class="mw-redirect" title="Locally convex">locally convex</a>) vector topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that are induced by linear functionals with distinct kernels will induce distinct vector topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></li> <li>However, while there are infinitely many vector topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim X\geq 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim X\geq 2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0aab2709b7dd0dfd9a8485fb1f2923f944737c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.15ex; height:2.509ex;" alt="{\displaystyle \dim X\geq 2,}"> there are, up to TVS-isomorphism, only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\dim X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\dim X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512a6ccef9974d940fe27e51cc4d8972a58e42ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.245ex; height:2.343ex;" alt="{\displaystyle 1+\dim X}"> vector topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> For instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:=\dim X=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>:=</mo> <mi>dim</mi> <mo>⁡</mo> <mi>X</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n:=\dim X=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7046d75c2d059e900ffdf62af0f5d7fadb6f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.643ex; height:2.176ex;" alt="{\displaystyle n:=\dim X=2}"> then the vector topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> are all TVS-isomorphic to one another.</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Non-vector_topologies">Non-vector topologies</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=16" title="Edit section: Non-vector topologies">edit</a>]</div> Discrete and cofinite topologies If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a non-trivial vector space (that is, of non-zero dimension) then the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (which is always <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable</a>) is not a TVS topology because despite making addition and negation continuous (which makes it into a <a href="/wiki/Topological_group" title="Topological group">topological group</a> under addition), it fails to make scalar multiplication continuous. The <a href="/wiki/Cofinite_topology" class="mw-redirect" title="Cofinite topology">cofinite topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (where a subset is open if and only if its complement is finite) is also not a TVS topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> <div class="mw-heading mw-heading2"><h2 id="Linear_maps">Linear maps</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=17" title="Edit section: Linear maps">edit</a>]</div> A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is continuous if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b884e2d65b3356219702968b6751485fb8f38570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}"> is bounded (as defined below) for some neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> of the origin. A <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> in a topological vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is either dense or closed. A <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> on a topological vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has either dense or closed kernel. Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is continuous if and only if its <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> is <a href="/wiki/Closed_set" title="Closed set">closed</a>. <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=18" title="Edit section: Types">edit</a>]</div> Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the <a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">closed graph theorem</a>, the <a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">open mapping theorem</a>, and the fact that the dual space of the space separates points in the space. Below are some common topological vector spaces, roughly in order of increasing "niceness." <ul><li><a href="/wiki/F-space" title="F-space">F-spaces</a> are <a href="/wiki/Complete_space" class="mw-redirect" title="Complete space">complete</a> topological vector spaces with a translation-invariant metric.<a href="#cite_note-FOOTNOTERudin19919_§1.8-31">[25]</a> These include <a href="/wiki/Lp_space" title="Lp space"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> spaces</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a4848bb7e353ef2eb87203e58dc3fa2f0a8546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p>0.}"></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector spaces</a>: here each point has a <a href="/wiki/Local_base" class="mw-redirect" title="Local base">local base</a> consisting of <a href="/wiki/Convex_set" title="Convex set">convex sets</a>.<a href="#cite_note-FOOTNOTERudin19919_§1.8-31">[25]</a> By a technique known as <a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski functionals</a> it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms.<a href="#cite_note-FOOTNOTERudin199127_Theorem_1.36-32">[26]</a> Local convexity is the minimum requirement for "geometrical" arguments like the <a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> spaces are locally convex (in fact, Banach spaces) for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\geq 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807af4aba28eab9b14fead4001f0bf1f2b93cbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p\geq 1,}"> but not for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f84eaa8b90c4e0ec96f4065fca8b29dea841a8b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.338ex; height:2.509ex;" alt="{\displaystyle 0<p<1.}"></li> <li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled spaces</a>: locally convex spaces where the <a href="/wiki/Banach%E2%80%93Steinhaus_theorem" class="mw-redirect" title="Banach–Steinhaus theorem">Banach–Steinhaus theorem</a> holds.</li> <li><a href="/wiki/Bornological_space" title="Bornological space">Bornological space</a>: a locally convex space where the <a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">continuous linear operators</a> to any locally convex space are exactly the <a href="/wiki/Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded linear operators</a>.</li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype space</a>: a locally convex space satisfying a variant of <a href="/wiki/Reflexive_space" title="Reflexive space">reflexivity condition</a>, where the dual space is endowed with the topology of uniform convergence on <a href="/wiki/Totally_bounded_space" title="Totally bounded space">totally bounded sets</a>.</li> <li><a href="/wiki/Montel_space" title="Montel space">Montel space</a>: a barrelled space where every <a href="/wiki/Closed_set" title="Closed set">closed</a> and <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">bounded set</a> is <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a>: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class -- <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c913ced284ad05bd42c8f004a3a83dee6323cb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} )}"> is a Fréchet space under the seminorms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \|f\|_{k,\ell }=\sup _{x\in [-k,k]}|f^{(\ell )}(x)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \|f\|_{k,\ell }=\sup _{x\in [-k,k]}|f^{(\ell )}(x)|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e49c32a6f2b210a53aa4bd19e23b7bfa0e48ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.052ex; height:3.509ex;" alt="{\textstyle \|f\|_{k,\ell }=\sup _{x\in [-k,k]}|f^{(\ell )}(x)|.}"> A locally convex F-space is a Fréchet space.<a href="#cite_note-FOOTNOTERudin19919_§1.8-31">[25]</a></li> <li><a href="/wiki/LF-space" title="LF-space">LF-spaces</a> are <a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">limits</a> of <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a>. <a href="/w/index.php?title=ILH_space&action=edit&redlink=1" class="new" title="ILH space (page does not exist)">ILH spaces</a> are <a href="/wiki/Inverse_limit" title="Inverse limit">inverse limits</a> of Hilbert spaces.</li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear spaces</a>: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a <a href="/wiki/Nuclear_operator" title="Nuclear operator">nuclear operator</a>.</li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed spaces</a> and <a href="/wiki/Seminormed_space" class="mw-redirect" title="Seminormed space">seminormed spaces</a>: locally convex spaces where the topology can be described by a single <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> or <a href="/w/index.php?title=Seminorm_(mathematics)&action=edit&redlink=1" class="new" title="Seminorm (mathematics) (page does not exist)">seminorm</a>. In normed spaces a linear operator is continuous if and only if it is bounded.</li> <li><a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>: Complete <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector spaces</a>. Most of functional analysis is formulated for Banach spaces. This class includes the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> spaces with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c206ad80b172eab6cfa550a466c62b47ac5043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty ,}"> the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BV}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e11b5fcd0ba19b882c2da7976d2fa8f275500a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.551ex; height:2.176ex;" alt="{\displaystyle BV}"> of <a href="/wiki/Bounded_variation" title="Bounded variation">functions of bounded variation</a>, and <a href="/wiki/Ba_space" title="Ba space">certain spaces</a> of measures.</li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive Banach spaces</a>: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is not reflexive is <a href="/wiki/Lp_space" title="Lp space"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}"></a>, whose dual is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"> but is strictly contained in the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7acbd1508221426c8deb0a177a71816205aa1f8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.105ex; height:2.343ex;" alt="{\displaystyle L^{\infty }.}"></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>: these have an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a>; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> spaces, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{2,k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{2,k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/138bdc9fa9bd33af5bd1a50416902d8ad04293e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.523ex; height:3.009ex;" alt="{\displaystyle W^{2,k},}"> and <a href="/wiki/Hardy_space" title="Hardy space">Hardy spaces</a>.</li> <li><a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"> with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"> there is only one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional topological vector space, up to isomorphism. It follows from this that any finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Dual_space">Dual space</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=19" title="Edit section: Dual space">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Algebraic_dual_space" class="mw-redirect" title="Algebraic dual space">Algebraic dual space</a>, <a href="/wiki/Continuous_dual_space" class="mw-redirect" title="Continuous dual space">Continuous dual space</a>, and <a href="/wiki/Strong_dual_space" title="Strong dual space">Strong dual space</a></div> Every topological vector space has a <a href="/wiki/Continuous_dual_space" class="mw-redirect" title="Continuous dual space">continuous dual space</a>—the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865f8505e90120a535a4ee68ca253dbd8ce7eb6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}"> of all continuous linear functionals, that is, <a href="/wiki/Continuous_linear_map" class="mw-redirect" title="Continuous linear map">continuous linear maps</a> from the space into the base field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a85c73bd0f34d264a8b2a361a14e66ab0231834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} .}"> A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'\to \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X'\to \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2113c34e9d6abdec24e986b3a2dc16bdf2b0126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.104ex; height:2.509ex;" alt="{\displaystyle X'\to \mathbb {K} }"> is continuous. This turns the dual into a locally convex topological vector space. This topology is called the <a href="/wiki/Weak_topology" title="Weak topology">weak-* topology</a>.<a href="#cite_note-FOOTNOTERudin199162-68_§3.8-3.14-33">[27]</a> This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see <a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu theorem</a>). Caution: Whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a non-normable locally convex space, then the pairing map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'\times X\to \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X'\times X\to \mathbb {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be02eb25aff3b66d7057fc29c13a2202521fecd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.924ex; height:2.509ex;" alt="{\displaystyle X'\times X\to \mathbb {K} }"> is never continuous, no matter which vector space topology one chooses on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X'.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc795cd053ac648fdaaf06eb968e57d0a91068a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.328ex; height:2.509ex;" alt="{\displaystyle X'.}"> A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.<a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-34">[28]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=20" title="Edit section: Properties">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Locally_convex_topological_vector_space#Properties" title="Locally convex topological vector space">Locally convex topological vector space § Properties</a></div> For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> the <a href="/wiki/Convex_set" title="Convex set">convex</a> (resp. <a href="/wiki/Balanced_set" title="Balanced set">balanced</a>, <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">disked</a>, closed convex, closed balanced, closed disked') hull of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is the smallest subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that has this property and contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}"> The closure (respectively, interior, <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a>, balanced hull, disked hull) of a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is sometimes denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e6af1f5b5dd8fc546cf94c9c2b9bfcaaba93d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.198ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}S}"> (respectively, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} _{X}S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} _{X}S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338928d071e43af1c2fb06e2a6dfca7f87aa657f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.202ex; height:2.509ex;" alt="{\displaystyle \operatorname {Int} _{X}S,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {co} S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {co} S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90221f1751c2a40ab80df0a64978ad8e5a25bdca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.728ex; height:2.509ex;" alt="{\displaystyle \operatorname {co} S,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {bal} S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>bal</mi> <mo>⁡</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {bal} S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0070ae6d2a3622f7d0304551f9c7db239345916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.635ex; height:2.509ex;" alt="{\displaystyle \operatorname {bal} S,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cobal} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cobal</mi> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cobal} S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d525f7d2c7ea96eb8b16d5bf5c9a616c0eb07e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.183ex; height:2.176ex;" alt="{\displaystyle \operatorname {cobal} S}">). The <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {co} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {co} S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e28196d9a7ba59064105fa148d156b2d4d837d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.081ex; height:2.176ex;" alt="{\displaystyle \operatorname {co} S}"> of a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is equal to the set of all <a href="/wiki/Convex_combination" title="Convex combination">convex combinations</a> of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0bcd8516b165aaacb234616d7d2d23478a35be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.146ex; height:2.509ex;" alt="{\displaystyle S,}"> which are finite <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}s_{1}+\cdots +t_{n}s_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}s_{1}+\cdots +t_{n}s_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465f9bd5a79c25cf4097641396d24c581f7e19a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.81ex; height:2.343ex;" alt="{\displaystyle t_{1}s_{1}+\cdots +t_{n}s_{n}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> is an integer, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1},\ldots ,s_{n}\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1},\ldots ,s_{n}\in S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3fbf6ca885ac8fb5fff9508e6579f5ce23b31f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.972ex; height:2.509ex;" alt="{\displaystyle s_{1},\ldots ,s_{n}\in S}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},\ldots ,t_{n}\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1},\ldots ,t_{n}\in [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985ff0e2b88aad029f13c1c1fa8f609f98788066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.623ex; height:2.843ex;" alt="{\displaystyle t_{1},\ldots ,t_{n}\in [0,1]}"> sum to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}"><a href="#cite_note-FOOTNOTERudin199138-35">[29]</a> The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it.<a href="#cite_note-FOOTNOTERudin199138-35">[29]</a> <div class="mw-heading mw-heading3"><h3 id="Neighborhoods_and_open_sets">Neighborhoods and open sets</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=21" title="Edit section: Neighborhoods and open sets">edit</a>]</div> Properties of neighborhoods and open sets Every TVS is <a href="/wiki/Connected_space" title="Connected space">connected</a><a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> and <a href="/wiki/Locally_connected_space" title="Locally connected space">locally connected</a><a href="#cite_note-FOOTNOTESchaeferWolff199935-36">[30]</a> and any connected open subset of a TVS is <a href="/wiki/Arcwise_connected" class="mw-redirect" title="Arcwise connected">arcwise connected</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf5e877bb3fa03f9e037f0b6c50a685536b99a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.122ex; height:2.343ex;" alt="{\displaystyle S+U}"> is an open set in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"><a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> has non-empty interior then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S-S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>−</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S-S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56405ef9fbae1faf971f3d9420c4e23a4902349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.839ex; height:2.343ex;" alt="{\displaystyle S-S}"> is a neighborhood of the origin.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> The open convex subsets of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (not necessarily Hausdorff or locally convex) are exactly those that are of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+\{x\in X:p(x)<1\}~=~\{x\in X:p(x-z)<1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+\{x\in X:p(x)<1\}~=~\{x\in X:p(x-z)<1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3255af7d16572daed485dac404352ba96258c80d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.08ex; height:2.843ex;" alt="{\displaystyle z+\{x\in X:p(x)<1\}~=~\{x\in X:p(x-z)<1\}}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff47a41328065a0e15e3a7089cfd392ee954e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.909ex; height:2.176ex;" alt="{\displaystyle z\in X}"> and some positive continuous <a href="/wiki/Sublinear_functional" class="mw-redirect" title="Sublinear functional">sublinear functional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"><a href="#cite_note-FOOTNOTENariciBeckenstein2011177–220-34">[28]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is an <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing</a> <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">disk</a> in a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:=p_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:=p_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1a0a2bc373801ad8994341d971db0a1e0ffc73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.867ex; height:2.009ex;" alt="{\displaystyle p:=p_{K}}"> is the <a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski functional</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> then<a href="#cite_note-FOOTNOTENariciBeckenstein2011119-120-37">[31]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} _{X}K~\subseteq ~\{x\in X:p(x)<1\}~\subseteq ~K~\subseteq ~\{x\in X:p(x)\leq 1\}~\subseteq ~\operatorname {cl} _{X}K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>K</mi> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>K</mi> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} _{X}K~\subseteq ~\{x\in X:p(x)<1\}~\subseteq ~K~\subseteq ~\{x\in X:p(x)\leq 1\}~\subseteq ~\operatorname {cl} _{X}K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00653569c46a4f8ba286ec23ff3c1c26bac2ac2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.342ex; height:2.843ex;" alt="{\displaystyle \operatorname {Int} _{X}K~\subseteq ~\{x\in X:p(x)<1\}~\subseteq ~K~\subseteq ~\{x\in X:p(x)\leq 1\}~\subseteq ~\operatorname {cl} _{X}K}"> where importantly, it was not assumed that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> had any topological properties nor that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> was continuous (which happens if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is a neighborhood of the origin). Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> be two vector topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \subseteq \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>⊆</mo> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau \subseteq \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b97a04eee3fda583d757c8bfe24b69257cc58fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.533ex; height:2.176ex;" alt="{\displaystyle \tau \subseteq \nu }"> if and only if whenever a net <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f06cbd9ae8f0977830b0571545f637115665c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.146ex; height:3.009ex;" alt="{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> converges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b155e12266a551b0b4958e6a80445588d2c5de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.055ex; height:2.843ex;" alt="{\displaystyle (X,\nu )}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{\bullet }\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\bullet }\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/937def0f95a8d85567d6557bb38ab5d59190d679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:7.16ex; height:2.509ex;" alt="{\displaystyle x_{\bullet }\to 0}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b42cadba3d621e09a00a4e133d83ad19f6d548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle (X,\tau ).}"><a href="#cite_note-FOOTNOTEWilansky201343-38">[32]</a> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"> be a neighborhood basis of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcee64dbb80f9474630b849a15567dc2acd95608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle S\subseteq X,}"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0deab6a01578b5b543b772df12dc0d2c593cc924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.797ex; height:2.176ex;" alt="{\displaystyle x\in X.}"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e320b55491aa18538e12df48c987694a2b66bc40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.368ex; height:2.509ex;" alt="{\displaystyle x\in \operatorname {cl} _{X}S}"> if and only if there exists a net <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\bullet }=\left(s_{N}\right)_{N\in {\mathcal {N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\bullet }=\left(s_{N}\right)_{N\in {\mathcal {N}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1884c08997fa0b6dc7a37e302241733f4defcef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.231ex; height:3.009ex;" alt="{\displaystyle s_{\bullet }=\left(s_{N}\right)_{N\in {\mathcal {N}}}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> (indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}">) such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\bullet }\to x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo stretchy="false">→</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\bullet }\to x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1dd7795792092a20bcd2b287d5903866d0a274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.26ex; margin-bottom: -0.412ex; width:7.088ex; height:2.176ex;" alt="{\displaystyle s_{\bullet }\to x}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"><a href="#cite_note-FOOTNOTEWilansky201342-39">[33]</a> This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a TVS that is of the <a href="/wiki/Second_category" class="mw-redirect" title="Second category">second category</a> in itself (that is, a <a href="/wiki/Nonmeager_space" class="mw-redirect" title="Nonmeager space">nonmeager space</a>) then any closed convex <a href="/wiki/Absorbing_set" title="Absorbing set">absorbing</a> subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a neighborhood of the origin.<a href="#cite_note-FOOTNOTERudin199155-40">[34]</a> This is no longer guaranteed if the set is not convex (a counter-example exists even in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7927b9ff2cf803f94f24dfc058375f09ab7c370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.811ex; height:2.676ex;" alt="{\displaystyle X=\mathbb {R} ^{2}}">) or if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is not of the second category in itself.<a href="#cite_note-FOOTNOTERudin199155-40">[34]</a> Interior If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4820d533f59b60d24c7aa2945bc48d5ae928856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.376ex; height:2.509ex;" alt="{\displaystyle R,S\subseteq X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> has non-empty interior then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} _{X}S~=~\operatorname {Int} _{X}\left(\operatorname {cl} _{X}S\right)~{\text{ and }}~\operatorname {cl} _{X}S~=~\operatorname {cl} _{X}\left(\operatorname {Int} _{X}S\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mtext> </mtext> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} _{X}S~=~\operatorname {Int} _{X}\left(\operatorname {cl} _{X}S\right)~{\text{ and }}~\operatorname {cl} _{X}S~=~\operatorname {cl} _{X}\left(\operatorname {Int} _{X}S\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49cbb1b7b418662e595e2405c9ce23e62f05bb8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.857ex; height:2.843ex;" alt="{\displaystyle \operatorname {Int} _{X}S~=~\operatorname {Int} _{X}\left(\operatorname {cl} _{X}S\right)~{\text{ and }}~\operatorname {cl} _{X}S~=~\operatorname {cl} _{X}\left(\operatorname {Int} _{X}S\right)}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} _{X}(R)+\operatorname {Int} _{X}(S)~\subseteq ~R+\operatorname {Int} _{X}S\subseteq \operatorname {Int} _{X}(R+S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>R</mi> <mo>+</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>⊆</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} _{X}(R)+\operatorname {Int} _{X}(S)~\subseteq ~R+\operatorname {Int} _{X}S\subseteq \operatorname {Int} _{X}(R+S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094d8ff68b2ebf18c4d529ac12c7cd6d08246eda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.807ex; height:2.843ex;" alt="{\displaystyle \operatorname {Int} _{X}(R)+\operatorname {Int} _{X}(S)~\subseteq ~R+\operatorname {Int} _{X}S\subseteq \operatorname {Int} _{X}(R+S).}"> The <a href="/wiki/Topological_interior" class="mw-redirect" title="Topological interior">topological interior</a> of a <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">disk</a> is not empty if and only if this interior contains the origin.<a href="#cite_note-FOOTNOTENariciBeckenstein2011108-41">[35]</a> More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is a <a href="/wiki/Balanced_set" title="Balanced set">balanced</a> set with non-empty interior <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} _{X}S\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} _{X}S\neq \varnothing }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ef0212f1502fd4f7c24cb73fca1257c8aa8d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.462ex; height:2.676ex;" alt="{\displaystyle \operatorname {Int} _{X}S\neq \varnothing }"> in a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}\cup \operatorname {Int} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}\cup \operatorname {Int} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf277316a2ebc484e353c74fd08a59bc590bf421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.625ex; height:2.843ex;" alt="{\displaystyle \{0\}\cup \operatorname {Int} _{X}S}"> will necessarily be balanced;<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> consequently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfae6d3a6bfd4827bc07984f20e50c7ec632fe52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.555ex; height:2.509ex;" alt="{\displaystyle \operatorname {Int} _{X}S}"> will be balanced if and only if it contains the origin.<a href="#cite_note-42">[proof 2]</a> For this (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \operatorname {Int} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \operatorname {Int} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8aed5c43f94081946a8982f566eb4ba256b64f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.559ex; height:2.509ex;" alt="{\displaystyle 0\in \operatorname {Int} _{X}S}">) to be true, it suffices for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> to also be convex (in addition to being balanced and having non-empty interior).;<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> The conclusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \operatorname {Int} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \operatorname {Int} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8aed5c43f94081946a8982f566eb4ba256b64f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.559ex; height:2.509ex;" alt="{\displaystyle 0\in \operatorname {Int} _{X}S}"> could be false if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is not also convex;<a href="#cite_note-FOOTNOTENariciBeckenstein2011108-41">[35]</a> for example, in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\mathbb {R} ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:=\mathbb {R} ^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c6084afc9a9122da0f6b1992ae65190326f71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.104ex; height:3.009ex;" alt="{\displaystyle X:=\mathbb {R} ^{2},}"> the interior of the closed and balanced set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:=\{(x,y):xy\geq 0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>x</mi> <mi>y</mi> <mo>≥</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S:=\{(x,y):xy\geq 0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4277fdeace12604cb95847edc8865cd32a4dd8aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.581ex; height:2.843ex;" alt="{\displaystyle S:=\{(x,y):xy\geq 0\}}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,y):xy>0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>x</mi> <mi>y</mi> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(x,y):xy>0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7eb40addd37704caf86a3e9825a280186ba805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.983ex; height:2.843ex;" alt="{\displaystyle \{(x,y):xy>0\}.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is convex and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<t\leq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>t</mi> <mo>≤</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<t\leq 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b263279253afcc9bbbed01297144b1a29d789ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.008ex; height:2.509ex;" alt="{\displaystyle 0<t\leq 1,}"> then<a href="#cite_note-FOOTNOTEJarchow1981101–104-43">[36]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\operatorname {Int} C+(1-t)\operatorname {cl} C~\subseteq ~\operatorname {Int} C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>Int</mi> <mo>⁡</mo> <mi>C</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cl</mi> <mo>⁡</mo> <mi>C</mi> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>Int</mi> <mo>⁡</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\operatorname {Int} C+(1-t)\operatorname {cl} C~\subseteq ~\operatorname {Int} C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f30b4d47a3f19457b7be9007cbaf03098d493f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.613ex; height:2.843ex;" alt="{\displaystyle t\operatorname {Int} C+(1-t)\operatorname {cl} C~\subseteq ~\operatorname {Int} C.}"> Explicitly, this means that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is a convex subset of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (not necessarily Hausdorff or locally convex), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \operatorname {int} _{X}C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <msub> <mi>int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \operatorname {int} _{X}C,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ad95e1bf997c99fbe222a6eb6965482eb66b2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.273ex; height:2.509ex;" alt="{\displaystyle y\in \operatorname {int} _{X}C,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {cl} _{X}C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \operatorname {cl} _{X}C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c37ac91d517ff60ef7565b84101846f1a6cbe16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.635ex; height:2.509ex;" alt="{\displaystyle x\in \operatorname {cl} _{X}C}"> then the open line segment joining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> belongs to the interior of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92402978f89ec5a7e982d699f86a9673769ab7d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle C;}"> that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{tx+(1-t)y:0<t<1\}\subseteq \operatorname {int} _{X}C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>:</mo> <mn>0</mn> <mo><</mo> <mi>t</mi> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <msub> <mi>int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{tx+(1-t)y:0<t<1\}\subseteq \operatorname {int} _{X}C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d939088ba7368a75f80fc043e12186967eadee2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.816ex; height:2.843ex;" alt="{\displaystyle \{tx+(1-t)y:0<t<1\}\subseteq \operatorname {int} _{X}C.}"><a href="#cite_note-FOOTNOTESchaeferWolff199938-44">[37]</a><a href="#cite_note-FOOTNOTEConway1990102-45">[38]</a><a href="#cite_note-46">[proof 3]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87353e8ceae2fe8f047c233b269f03d0cb80da93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.142ex; height:2.343ex;" alt="{\displaystyle N\subseteq X}"> is any balanced neighborhood of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Int} _{X}N\subseteq B_{1}N=\bigcup _{0<|a|<1}aN\subseteq N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mo>⊆</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>N</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mrow> </munder> <mi>a</mi> <mi>N</mi> <mo>⊆</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {Int} _{X}N\subseteq B_{1}N=\bigcup _{0<|a|<1}aN\subseteq N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f16b86cbd37a52bf5ad87d129ab3e157d8402b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.195ex; height:3.343ex;" alt="{\textstyle \operatorname {Int} _{X}N\subseteq B_{1}N=\bigcup _{0<|a|<1}aN\subseteq N}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa091eb428443c9c5c5fcf32a69d3665c89e00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{1}}"> is the set of all scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|<1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dbe08091cffcb1569c9a9f4d4ab3feccfc2c9a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.431ex; height:2.843ex;" alt="{\displaystyle |a|<1.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> belongs to the interior of a convex set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \operatorname {cl} _{X}S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \operatorname {cl} _{X}S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/032dc87f0f21e35c9536525e2f118eb6432ed110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.841ex; height:2.509ex;" alt="{\displaystyle y\in \operatorname {cl} _{X}S,}"> then the half-open line segment <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,y):=\{tx+(1-t)y:0<t\leq 1\}\subseteq \operatorname {Int} _{X}{\text{ if }}x\neq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>:</mo> <mn>0</mn> <mo><</mo> <mi>t</mi> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y):=\{tx+(1-t)y:0<t\leq 1\}\subseteq \operatorname {Int} _{X}{\text{ if }}x\neq y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde66e53cf718dce89f3502e3954d8da0882ff53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.515ex; height:2.843ex;" alt="{\displaystyle [x,y):=\{tx+(1-t)y:0<t\leq 1\}\subseteq \operatorname {Int} _{X}{\text{ if }}x\neq y}"> and<a href="#cite_note-FOOTNOTESchaeferWolff199938-44">[37]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,x)=\varnothing {\text{ if }}x=y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">∅</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,x)=\varnothing {\text{ if }}x=y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3298cf02c0c323ab54963f5cfc5cfc0393a19b70" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.902ex; height:2.843ex;" alt="{\displaystyle [x,x)=\varnothing {\text{ if }}x=y.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> is a <a href="/wiki/Balanced_set" title="Balanced set">balanced</a> neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}:=\{a\in \mathbb {K} :|a|<1\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}:=\{a\in \mathbb {K} :|a|<1\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89eee287b4fba56054d671c31a7f931d9e104d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.135ex; height:2.843ex;" alt="{\displaystyle B_{1}:=\{a\in \mathbb {K} :|a|<1\},}"> then by considering intersections of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\cap \mathbb {R} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\cap \mathbb {R} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/238dfa1b294bc157c8388a158dc195a20ed67bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.654ex; height:2.176ex;" alt="{\displaystyle N\cap \mathbb {R} x}"> (which are convex <a href="/wiki/Symmetric_set" title="Symmetric set">symmetric</a> neighborhoods of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> in the real TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa96e75c02660adcf2c26329b08f7d34dc5d164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.008ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} x}">) it follows that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} N=[0,1)\operatorname {Int} N=(-1,1)N=B_{1}N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Int</mi> <mo>⁡</mo> <mi>N</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>Int</mi> <mo>⁡</mo> <mi>N</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>N</mi> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} N=[0,1)\operatorname {Int} N=(-1,1)N=B_{1}N,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bed2625fc0c0bc498b6a4f3e85a433025bf47ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.136ex; height:2.843ex;" alt="{\displaystyle \operatorname {Int} N=[0,1)\operatorname {Int} N=(-1,1)N=B_{1}N,}"> and furthermore, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {Int} N{\text{ and }}r:=\sup\{r>0:[0,r)x\subseteq N\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>Int</mi> <mo>⁡</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>r</mi> <mo>:=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mi>r</mi> <mo>></mo> <mn>0</mn> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>⊆</mo> <mi>N</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \operatorname {Int} N{\text{ and }}r:=\sup\{r>0:[0,r)x\subseteq N\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504e84589fa4ec71fde5e4a0a344040c62ddce66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.722ex; height:2.843ex;" alt="{\displaystyle x\in \operatorname {Int} N{\text{ and }}r:=\sup\{r>0:[0,r)x\subseteq N\}}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>1{\text{ and }}[0,r)x\subseteq \operatorname {Int} N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>⊆</mo> <mi>Int</mi> <mo>⁡</mo> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>1{\text{ and }}[0,r)x\subseteq \operatorname {Int} N,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef51922dd15d1dd1104c787baaef857f973bbcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.577ex; height:2.843ex;" alt="{\displaystyle r>1{\text{ and }}[0,r)x\subseteq \operatorname {Int} N,}"> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\neq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≠</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\neq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b49ea8cab98f23b4cf4148059ba6f1be583669" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.471ex; height:2.676ex;" alt="{\displaystyle r\neq \infty }"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rx\in \operatorname {cl} N\setminus \operatorname {Int} N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>x</mi> <mo>∈</mo> <mi>cl</mi> <mo>⁡</mo> <mi>N</mi> <mo class="MJX-variant">∖</mo> <mi>Int</mi> <mo>⁡</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rx\in \operatorname {cl} N\setminus \operatorname {Int} N.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2ac1e082468c348e52047d799b7c8f991c068f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.678ex; height:2.843ex;" alt="{\displaystyle rx\in \operatorname {cl} N\setminus \operatorname {Int} N.}"> <div class="mw-heading mw-heading3"><h3 id="Non-Hausdorff_spaces_and_the_closure_of_the_origin">Non-Hausdorff spaces and the closure of the origin</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=22" title="Edit section: Non-Hausdorff spaces and the closure of the origin">edit</a>]</div> A topological vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is Hausdorff if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"> is a closed subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> or equivalently, if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}=\operatorname {cl} _{X}\{0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}=\operatorname {cl} _{X}\{0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb50480f44278b909bd36bae583a6bf2eef617c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.032ex; height:2.843ex;" alt="{\displaystyle \{0\}=\operatorname {cl} _{X}\{0\}.}"> Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"> is a vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> the same is true of its closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d0a68f47679cff5e8f92c7342744144e4bd16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.446ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\},}"> which is referred to as the closure of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> This vector space satisfies <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}=\bigcap _{N\in {\mathcal {N}}(0)}N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </munder> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}=\bigcap _{N\in {\mathcal {N}}(0)}N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4733c6a3fb9127324fdd4fee9bb47429931caeb6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:18.614ex; height:6.176ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}=\bigcap _{N\in {\mathcal {N}}(0)}N}"> so that in particular, every neighborhood of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> contains the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> as a subset. The <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> is always the <a href="/wiki/Trivial_topology" title="Trivial topology">trivial topology</a>, which in particular implies that the topological vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> a <a href="/wiki/Compact_space" title="Compact space">compact space</a> (even if its dimension is non-zero or even infinite) and consequently also a <a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">bounded subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5092383232dea9773e6f99baaa92bb31a25f1b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \{0\}.}"><a href="#cite_note-FOOTNOTENariciBeckenstein2011155–176-17">[14]</a> Every subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> also carries the trivial topology and so is itself a compact, and thus also complete, <a href="/wiki/Topological_subspace" class="mw-redirect" title="Topological subspace">subspace</a> (see footnote for a proof).<a href="#cite_note-47">[proof 4]</a> In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is not Hausdorff then there exist subsets that are both compact and complete but not closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">;<a href="#cite_note-FOOTNOTENariciBeckenstein201147–66-48">[39]</a> for instance, this will be true of any non-empty proper subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b279050c0f1b53818d9c4bebebbf24bf37017d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.446ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> is compact, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}S=S+\operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>=</mo> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}S=S+\operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6fd1c4ba0e732b64d9f1c292fddfe387d5ef47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.435ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}S=S+\operatorname {cl} _{X}\{0\}}"> and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are <a href="/wiki/Relatively_compact" class="mw-redirect" title="Relatively compact">relatively compact</a>),<a href="#cite_note-FOOTNOTENariciBeckenstein2011156-49">[40]</a> which is not guaranteed for arbitrary non-Hausdorff <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>.<a href="#cite_note-50">[note 6]</a> For every subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcee64dbb80f9474630b849a15567dc2acd95608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle S\subseteq X,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbe13b6a9a4f6c94ebd3f045169027384f65d44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.435ex; height:2.843ex;" alt="{\displaystyle S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S}"> and consequently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> is open or closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f081cd80c08b1eaa80d8be118a6bdc47b038bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.736ex; height:2.843ex;" alt="{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}"><a href="#cite_note-ProofSumOfSetAndClosureOf0-51">[proof 5]</a> (so that this arbitrary open or closed subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> can be described as a <a href="/wiki/Tube_lemma" title="Tube lemma">"tube"</a> whose vertical side is the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}">). For any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> of this TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> the following are equivalent: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is <a href="/wiki/Totally_bounded_space" title="Totally bounded space">totally bounded</a>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+\operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+\operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eff267485c77ceb0bc109ed67a09e204c3915ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.139ex; height:2.843ex;" alt="{\displaystyle S+\operatorname {cl} _{X}\{0\}}"> is totally bounded.<a href="#cite_note-FOOTNOTESchaeferWolff199912–35-52">[41]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e6af1f5b5dd8fc546cf94c9c2b9bfcaaba93d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.198ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}S}"> is totally bounded.<a href="#cite_note-FOOTNOTESchaeferWolff199925-53">[42]</a><a href="#cite_note-FOOTNOTEJarchow198156–73-54">[43]</a></li> <li>The image if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> under the canonical quotient map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47617a62510be2c29b955cec78ce6045ceaadbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.732ex; height:2.843ex;" alt="{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}"> is totally bounded.<a href="#cite_note-FOOTNOTESchaeferWolff199912–35-52">[41]</a></li></ul> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a vector subspace of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aaa4313768efcc32d3d5a1de282d7bbe08baec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.585ex; height:2.843ex;" alt="{\displaystyle X/M}"> is Hausdorff if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> Moreover, the <a href="/wiki/Quotient_map" class="mw-redirect" title="Quotient map">quotient map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q:X\to X/\operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q:X\to X/\operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd04e9efb5e6cceef72545b1c8993f22bc12e85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.929ex; height:2.843ex;" alt="{\displaystyle q:X\to X/\operatorname {cl} _{X}\{0\}}"> is always a <a href="/wiki/Open_and_closed_maps" title="Open and closed maps">closed map</a> onto the (necessarily) Hausdorff TVS.<a href="#cite_note-FOOTNOTENariciBeckenstein2011107–112-55">[44]</a> Every vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that is an algebraic complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> (that is, a vector subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}=H\cap \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>H</mi> <mo>∩</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}=H\cap \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e55d94661cf37853defe3c73899ff569968e4b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.031ex; height:2.843ex;" alt="{\displaystyle \{0\}=H\cap \operatorname {cl} _{X}\{0\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=H+\operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>H</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=H+\operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e647f488fe6578c54cfd77e1b0fb4db23a3530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.781ex; height:2.843ex;" alt="{\displaystyle X=H+\operatorname {cl} _{X}\{0\}}">) is a <a href="/wiki/Complemented_subspace" title="Complemented subspace">topological complement</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b279050c0f1b53818d9c4bebebbf24bf37017d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.446ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}.}"> Consequently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> is an algebraic complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then the addition map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\times \operatorname {cl} _{X}\{0\}\to X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>×</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\times \operatorname {cl} _{X}\{0\}\to X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424bae1e5c4a15ae058f93c2b03855d80b866b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.944ex; height:2.843ex;" alt="{\displaystyle H\times \operatorname {cl} _{X}\{0\}\to X,}"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h,n)\mapsto h+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>h</mi> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h,n)\mapsto h+n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af42ff2b00b7c7bd1f30e752cb2cc1b42b62e7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.765ex; height:2.843ex;" alt="{\displaystyle (h,n)\mapsto h+n}"> is a TVS-isomorphism, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> is necessarily Hausdorff and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> has the <a href="/wiki/Indiscrete_topology" class="mw-redirect" title="Indiscrete topology">indiscrete topology</a>.<a href="#cite_note-FOOTNOTEWilansky201363-56">[45]</a> Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is a Hausdorff <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">completion</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\times \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>×</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\times \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cceeb7b546dd24ad1603d376652ccd7042c74a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.406ex; height:2.843ex;" alt="{\displaystyle C\times \operatorname {cl} _{X}\{0\}}"> is a completion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\cong H\times \operatorname {cl} _{X}\{0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>≅</mo> <mi>H</mi> <mo>×</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\cong H\times \operatorname {cl} _{X}\{0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beaf5177f4bcbc087f31f200573c02dd6c9e0c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.428ex; height:2.843ex;" alt="{\displaystyle X\cong H\times \operatorname {cl} _{X}\{0\}.}"><a href="#cite_note-FOOTNOTESchaeferWolff199912–35-52">[41]</a> <div class="mw-heading mw-heading3"><h3 id="Closed_and_compact_sets">Closed and compact sets</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=23" title="Edit section: Closed and compact sets">edit</a>]</div> Compact and totally bounded sets A subset of a TVS is compact if and only if it is complete and <a href="/wiki/Totally_bounded_space" title="Totally bounded space">totally bounded</a>.<a href="#cite_note-FOOTNOTENariciBeckenstein201147–66-48">[39]</a> Thus, in a <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete topological vector space</a>, a closed and totally bounded subset is compact.<a href="#cite_note-FOOTNOTENariciBeckenstein201147–66-48">[39]</a> A subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is <a href="/wiki/Totally_bounded_space" title="Totally bounded space">totally bounded</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e6af1f5b5dd8fc546cf94c9c2b9bfcaaba93d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.198ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}S}"> is totally bounded,<a href="#cite_note-FOOTNOTESchaeferWolff199925-53">[42]</a><a href="#cite_note-FOOTNOTEJarchow198156–73-54">[43]</a> if and only if its image under the canonical quotient map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47617a62510be2c29b955cec78ce6045ceaadbc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.732ex; height:2.843ex;" alt="{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}"> is totally bounded.<a href="#cite_note-FOOTNOTESchaeferWolff199912–35-52">[41]</a> Every relatively compact set is totally bounded<a href="#cite_note-FOOTNOTENariciBeckenstein201147–66-48">[39]</a> and the closure of a totally bounded set is totally bounded.<a href="#cite_note-FOOTNOTENariciBeckenstein201147–66-48">[39]</a> The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded.<a href="#cite_note-FOOTNOTENariciBeckenstein201147–66-48">[39]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is a subset of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> such that every sequence in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> has a cluster point in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is totally bounded.<a href="#cite_note-FOOTNOTESchaeferWolff199912–35-52">[41]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is a compact subset of a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fea4b5006afa0eb81c21b577efe304ab545d571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}"> then there exists a neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> of 0 such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K+N\subseteq U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>+</mo> <mi>N</mi> <mo>⊆</mo> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K+N\subseteq U.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3012d2139b859f69c2ca29652bfeadae41081ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.498ex; height:2.343ex;" alt="{\displaystyle K+N\subseteq U.}"><a href="#cite_note-FOOTNOTENariciBeckenstein201119–45-57">[46]</a> Closure and closed set The closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a <a href="/wiki/Barrelled_space#barrel" title="Barrelled space">barrel</a>. The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> is a closed neighborhood of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∩</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cap N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56643414a96610663ee313767b4408826852931a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.429ex; height:2.176ex;" alt="{\displaystyle U\cap N}"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"><a href="#cite_note-FOOTNOTENariciBeckenstein201119–45-57">[46]</a> The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> (see this footnote<a href="#cite_note-58">[note 7]</a> for examples). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> is a scalar then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\operatorname {cl} _{X}S\subseteq \operatorname {cl} _{X}(aS),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>⊆</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>S</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\operatorname {cl} _{X}S\subseteq \operatorname {cl} _{X}(aS),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f3da4ecbaf69fa69499a50bdcb32b9f764c585" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.41ex; height:2.843ex;" alt="{\displaystyle a\operatorname {cl} _{X}S\subseteq \operatorname {cl} _{X}(aS),}"> where if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is Hausdorff, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0,{\text{ or }}S=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>S</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0,{\text{ or }}S=\varnothing }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ade3f675109d75e99033b328000ea5a2992a31e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.166ex; height:2.676ex;" alt="{\displaystyle a\neq 0,{\text{ or }}S=\varnothing }"> then equality holds: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(aS)=a\operatorname {cl} _{X}S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(aS)=a\operatorname {cl} _{X}S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d77b7ca4eee7e831d7cfeffef35d2dec892c5f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.41ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(aS)=a\operatorname {cl} _{X}S.}"> In particular, every non-zero scalar multiple of a closed set is closed. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is a set of scalars such that neither <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} S{\text{ nor }}\operatorname {cl} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cl</mi> <mo>⁡</mo> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> nor </mtext> </mrow> <mi>cl</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} S{\text{ nor }}\operatorname {cl} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbbe58e6042d36dbfb9abc55de036fba65f24fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.29ex; height:2.176ex;" alt="{\displaystyle \operatorname {cl} S{\text{ nor }}\operatorname {cl} A}"> contain zero then<a href="#cite_note-FOOTNOTEWilansky201343–44-59">[47]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\operatorname {cl} A\right)\left(\operatorname {cl} _{X}S\right)=\operatorname {cl} _{X}(AS).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>cl</mi> <mo>⁡</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\operatorname {cl} A\right)\left(\operatorname {cl} _{X}S\right)=\operatorname {cl} _{X}(AS).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8443c880c9e923c9f7c9a55a99564839cb710e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.122ex; height:2.843ex;" alt="{\displaystyle \left(\operatorname {cl} A\right)\left(\operatorname {cl} _{X}S\right)=\operatorname {cl} _{X}(AS).}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X{\text{ and }}S+S\subseteq 2\operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>S</mi> <mo>+</mo> <mi>S</mi> <mo>⊆</mo> <mn>2</mn> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X{\text{ and }}S+S\subseteq 2\operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9ea9e119cc5ab975cf40bd271e33707ffda8b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.171ex; height:2.509ex;" alt="{\displaystyle S\subseteq X{\text{ and }}S+S\subseteq 2\operatorname {cl} _{X}S}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e6af1f5b5dd8fc546cf94c9c2b9bfcaaba93d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.198ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}S}"> is convex.<a href="#cite_note-FOOTNOTEWilansky201343–44-59">[47]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4820d533f59b60d24c7aa2945bc48d5ae928856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.376ex; height:2.509ex;" alt="{\displaystyle R,S\subseteq X}"> then<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)~\subseteq ~\operatorname {cl} _{X}(R+S)~{\text{ and }}~\operatorname {cl} _{X}\left[\operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)\right]~=~\operatorname {cl} _{X}(R+S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mtext> </mtext> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)~\subseteq ~\operatorname {cl} _{X}(R+S)~{\text{ and }}~\operatorname {cl} _{X}\left[\operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)\right]~=~\operatorname {cl} _{X}(R+S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796f38a799f8ecde14d825ebfb3c0ac0a61b77c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:75.496ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)~\subseteq ~\operatorname {cl} _{X}(R+S)~{\text{ and }}~\operatorname {cl} _{X}\left[\operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)\right]~=~\operatorname {cl} _{X}(R+S)}"> and so consequently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R+S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c197ee381f4ea9804e5d5b33a0de0997fc99f722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.104ex; height:2.343ex;" alt="{\displaystyle R+S}"> is closed then so is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6211c33e0b5676859bbcda3ffbde594fb3cabf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.992ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S).}"><a href="#cite_note-FOOTNOTEWilansky201343–44-59">[47]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a real TVS and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcee64dbb80f9474630b849a15567dc2acd95608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle S\subseteq X,}"> then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigcap _{r>1}rS\subseteq \operatorname {cl} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>></mo> <mn>1</mn> </mrow> </munder> <mi>r</mi> <mi>S</mi> <mo>⊆</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcap _{r>1}rS\subseteq \operatorname {cl} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dce97fbcbd7ef280dbe69b4299209a8e6b774d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.073ex; height:5.509ex;" alt="{\displaystyle \bigcap _{r>1}rS\subseteq \operatorname {cl} _{X}S}"> where the left hand side is independent of the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/382fc9d9db960b232f5960d73b4a4762c7a047e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X;}"> moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is a convex neighborhood of the origin then equality holds. For any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcee64dbb80f9474630b849a15567dc2acd95608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle S\subseteq X,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}S~=~\bigcap _{N\in {\mathcal {N}}}(S+N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mrow> </munder> <mo stretchy="false">(</mo> <mi>S</mi> <mo>+</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}S~=~\bigcap _{N\in {\mathcal {N}}}(S+N)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98eb2175556aa4666e8f073fc2e497edfc56d1bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.221ex; height:5.843ex;" alt="{\displaystyle \operatorname {cl} _{X}S~=~\bigcap _{N\in {\mathcal {N}}}(S+N)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"> is any neighborhood basis at the origin for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"><a href="#cite_note-FOOTNOTENariciBeckenstein201180-60">[48]</a> However, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}U~\supseteq ~\bigcap \{U:S\subseteq U,U{\text{ is open in }}X\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>U</mi> <mtext> </mtext> <mo>⊇</mo> <mtext> </mtext> <mo>⋂</mo> <mo fence="false" stretchy="false">{</mo> <mi>U</mi> <mo>:</mo> <mi>S</mi> <mo>⊆</mo> <mi>U</mi> <mo>,</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is open in </mtext> </mrow> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}U~\supseteq ~\bigcap \{U:S\subseteq U,U{\text{ is open in }}X\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80bb853c202f17e005a844fa73d6a33cf4f986ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:40.536ex; height:3.843ex;" alt="{\displaystyle \operatorname {cl} _{X}U~\supseteq ~\bigcap \{U:S\subseteq U,U{\text{ is open in }}X\}}"> and it is possible for this containment to be proper<a href="#cite_note-FOOTNOTENariciBeckenstein2011108–109-61">[49]</a> (for example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2e2b6427cd2b517be352b378a1830c1540e3a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.757ex; height:2.176ex;" alt="{\displaystyle X=\mathbb {R} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is the rational numbers). It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}U\subseteq U+U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>U</mi> <mo>⊆</mo> <mi>U</mi> <mo>+</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}U\subseteq U+U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c65424d5beb0cd6ec80f0b2ebc39e4df1518108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.985ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{X}U\subseteq U+U}"> for every neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"><a href="#cite_note-FOOTNOTEJarchow198130–32-62">[50]</a> Closed hulls In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.<a href="#cite_note-FOOTNOTENariciBeckenstein2011155–176-17">[14]</a> <ul><li>The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10290c37a0aaba09afe9bf7f39669e91b3990206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.849ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S).}"><a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a></li> <li>The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(\operatorname {bal} S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>bal</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(\operatorname {bal} S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c12248499a39a90582d790347a94a024c43862" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.756ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(\operatorname {bal} S).}"><a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a></li> <li>The closed <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">disked</a> hull of a set is equal to the closure of the disked hull of that set; that is, equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(\operatorname {cobal} S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cobal</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(\operatorname {cobal} S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/035ad389a1f4b1f9c22dc792f056a936bafe3c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.951ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(\operatorname {cobal} S).}"><a href="#cite_note-FOOTNOTENariciBeckenstein2011109-63">[51]</a></li></ul> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4820d533f59b60d24c7aa2945bc48d5ae928856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.376ex; height:2.509ex;" alt="{\displaystyle R,S\subseteq X}"> and the closed convex hull of one of the sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is compact then<a href="#cite_note-FOOTNOTENariciBeckenstein2011109-63">[51]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R+S))~=~\operatorname {cl} _{X}(\operatorname {co} R)+\operatorname {cl} _{X}(\operatorname {co} S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R+S))~=~\operatorname {cl} _{X}(\operatorname {co} R)+\operatorname {cl} _{X}(\operatorname {co} S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b58c7fd470fd58c353af0b7277586ba190c3c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.032ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R+S))~=~\operatorname {cl} _{X}(\operatorname {co} R)+\operatorname {cl} _{X}(\operatorname {co} S).}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4820d533f59b60d24c7aa2945bc48d5ae928856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.376ex; height:2.509ex;" alt="{\displaystyle R,S\subseteq X}"> each have a closed convex hull that is compact (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(\operatorname {co} R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c52428800596eb83bab219e0082ab0056543d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.467ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} R)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b3ea8f027c083968e9072560b3ae154bcbe151" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.202ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S)}"> are compact) then<a href="#cite_note-FOOTNOTENariciBeckenstein2011109-63">[51]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R\cup S))~=~\operatorname {co} \left[\operatorname {cl} _{X}(\operatorname {co} R)\cup \operatorname {cl} _{X}(\operatorname {co} S)\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>∪</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>co</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R\cup S))~=~\operatorname {co} \left[\operatorname {cl} _{X}(\operatorname {co} R)\cup \operatorname {cl} _{X}(\operatorname {co} S)\right].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e41b11bfc0ed54d2de2503e9df59ee85e3ce25d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.005ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R\cup S))~=~\operatorname {co} \left[\operatorname {cl} _{X}(\operatorname {co} R)\cup \operatorname {cl} _{X}(\operatorname {co} S)\right].}"> Hulls and compactness In a general TVS, the closed convex hull of a compact set may fail to be compact. The balanced hull of a compact (respectively, <a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a>) set has that same property.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> The convex hull of a finite union of compact convex sets is again compact and convex.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Other_properties">Other properties</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=24" title="Edit section: Other properties">edit</a>]</div> Meager, nowhere dense, and Baire A <a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">disk</a> in a TVS is not <a href="/wiki/Nowhere_dense" class="mw-redirect" title="Nowhere dense">nowhere dense</a> if and only if its closure is a neighborhood of the origin.<a href="#cite_note-FOOTNOTENariciBeckenstein2011371–423-9">[9]</a> A vector subspace of a TVS that is closed but not open is <a href="/wiki/Nowhere_dense" class="mw-redirect" title="Nowhere dense">nowhere dense</a>.<a href="#cite_note-FOOTNOTENariciBeckenstein2011371–423-9">[9]</a> Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a TVS that does not carry the <a href="/wiki/Indiscrete_topology" class="mw-redirect" title="Indiscrete topology">indiscrete topology</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a <a href="/wiki/Baire_space" title="Baire space">Baire space</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has no balanced absorbing nowhere dense subset.<a href="#cite_note-FOOTNOTENariciBeckenstein2011371–423-9">[9]</a> A TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a Baire space if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is <a href="/wiki/Nonmeager" class="mw-redirect" title="Nonmeager">nonmeager</a>, which happens if and only if there does not exist a <a href="/wiki/Nowhere_dense" class="mw-redirect" title="Nowhere dense">nowhere dense</a> set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X=\bigcup _{n\in \mathbb {N} }nD.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mi>n</mi> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X=\bigcup _{n\in \mathbb {N} }nD.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d965f9bf5cdc6071da0a52c752ebc5413872a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.869ex; height:3.009ex;" alt="{\textstyle X=\bigcup _{n\in \mathbb {N} }nD.}"><a href="#cite_note-FOOTNOTENariciBeckenstein2011371–423-9">[9]</a> Every <a href="/wiki/Nonmeager" class="mw-redirect" title="Nonmeager">nonmeager</a> locally convex TVS is a <a href="/wiki/Barrelled_space" title="Barrelled space">barrelled space</a>.<a href="#cite_note-FOOTNOTENariciBeckenstein2011371–423-9">[9]</a> Important algebraic facts and common misconceptions If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2S\subseteq S+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>S</mi> <mo>⊆</mo> <mi>S</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2S\subseteq S+S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957be222fb4c2fba5e823d26b6064690da9b7257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.599ex; height:2.343ex;" alt="{\displaystyle 2S\subseteq S+S}">; if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is convex then equality holds. For an example where equality does not hold, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> be non-zero and set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{-x,x\};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{-x,x\};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009dbacf5ba1d3c5af50e4bb9b1117213414660d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.071ex; height:2.843ex;" alt="{\displaystyle S=\{-x,x\};}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{x,2x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mn>2</mn> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{x,2x\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d31ec17aa3bf0efbdca1a9a292259736880a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.778ex; height:2.843ex;" alt="{\displaystyle S=\{x,2x\}}"> also works. A subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is convex if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s+t)C=sC+tC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>C</mi> <mo>=</mo> <mi>s</mi> <mi>C</mi> <mo>+</mo> <mi>t</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s+t)C=sC+tC}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f48666bcd73a47410026ac96eb0ecd3bd25e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.748ex; height:2.843ex;" alt="{\displaystyle (s+t)C=sC+tC}"> for all positive real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s>0{\text{ and }}t>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>></mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s>0{\text{ and }}t>0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed15769527075bccd09fc687edafb7d41398997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.008ex; height:2.509ex;" alt="{\displaystyle s>0{\text{ and }}t>0,}"><a href="#cite_note-FOOTNOTERudin199138-35">[29]</a> or equivalently, if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tC+(1-t)C\subseteq C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>C</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>C</mi> <mo>⊆</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tC+(1-t)C\subseteq C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb599adcd27c817309d8fbb12071d50533a33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.729ex; height:2.843ex;" alt="{\displaystyle tC+(1-t)C\subseteq C}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq t\leq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq t\leq 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de2f7f2acd36689fdbad7f715c93b7c69006fbcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.008ex; height:2.343ex;" alt="{\displaystyle 0\leq t\leq 1.}"><a href="#cite_note-FOOTNOTERudin19916-64">[52]</a> The <a href="/wiki/Convex_balanced_hull" class="mw-redirect" title="Convex balanced hull">convex balanced hull</a> of a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"> is equal to the convex hull of the <a href="/wiki/Balanced_hull" class="mw-redirect" title="Balanced hull">balanced hull</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e269413adacab44c96c066d297274b4163ec5035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.146ex; height:2.509ex;" alt="{\displaystyle S;}"> that is, it is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {co} (\operatorname {bal} S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>bal</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {co} (\operatorname {bal} S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0905c537819f186924d4d5d05e96ffb2018a25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.639ex; height:2.843ex;" alt="{\displaystyle \operatorname {co} (\operatorname {bal} S).}"> But in general, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {bal} (\operatorname {co} S)~\subseteq ~\operatorname {cobal} S~=~\operatorname {co} (\operatorname {bal} S),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>bal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>⊆</mo> <mtext> </mtext> <mi>cobal</mi> <mo>⁡</mo> <mi>S</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>bal</mi> <mo>⁡</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {bal} (\operatorname {co} S)~\subseteq ~\operatorname {cobal} S~=~\operatorname {co} (\operatorname {bal} S),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67552ef0bd8b237d531b055b636c4eb685cfa9d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.108ex; height:2.843ex;" alt="{\displaystyle \operatorname {bal} (\operatorname {co} S)~\subseteq ~\operatorname {cobal} S~=~\operatorname {co} (\operatorname {bal} S),}"> where the inclusion might be strict since the <a href="/wiki/Balanced_hull" class="mw-redirect" title="Balanced hull">balanced hull</a> of a convex set need not be convex (counter-examples exist even in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}">). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4820d533f59b60d24c7aa2945bc48d5ae928856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.376ex; height:2.509ex;" alt="{\displaystyle R,S\subseteq X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> is a scalar then<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(R+S)=aR+aS,~{\text{ and }}~\operatorname {co} (R+S)=\operatorname {co} R+\operatorname {co} S,~{\text{ and }}~\operatorname {co} (aS)=a\operatorname {co} S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>R</mi> <mo>+</mo> <mi>a</mi> <mi>S</mi> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mtext> </mtext> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>co</mi> <mo>⁡</mo> <mi>R</mi> <mo>+</mo> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mtext> </mtext> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>co</mi> <mo>⁡</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(R+S)=aR+aS,~{\text{ and }}~\operatorname {co} (R+S)=\operatorname {co} R+\operatorname {co} S,~{\text{ and }}~\operatorname {co} (aS)=a\operatorname {co} S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d36cd0682e07cec02d58db5cbc6a78f2f16cb9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.437ex; height:2.843ex;" alt="{\displaystyle a(R+S)=aR+aS,~{\text{ and }}~\operatorname {co} (R+S)=\operatorname {co} R+\operatorname {co} S,~{\text{ and }}~\operatorname {co} (aS)=a\operatorname {co} S.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,S\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4820d533f59b60d24c7aa2945bc48d5ae928856e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.376ex; height:2.509ex;" alt="{\displaystyle R,S\subseteq X}"> are convex non-empty disjoint sets and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\not \in R\cup S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>R</mi> <mo>∪</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in R\cup S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89652f70c28d7b56d766fea69e01535d31922a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.663ex; height:2.676ex;" alt="{\displaystyle x\not \in R\cup S,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cap \operatorname {co} (R\cup \{x\})=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∩</mo> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\cap \operatorname {co} (R\cup \{x\})=\varnothing }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e08b9f3e4288088eef313e173de4dfbc86f657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.994ex; height:2.843ex;" alt="{\displaystyle S\cap \operatorname {co} (R\cup \{x\})=\varnothing }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cap \operatorname {co} (S\cup \{x\})=\varnothing .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∩</mo> <mi>co</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">∅</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cap \operatorname {co} (S\cup \{x\})=\varnothing .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73ec9c026d962d66ce1c37e6be2dc4c04d8913f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.641ex; height:2.843ex;" alt="{\displaystyle R\cap \operatorname {co} (S\cup \{x\})=\varnothing .}"> In any non-trivial vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"> there exist two disjoint non-empty convex subsets whose union is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> Other properties Every TVS topology can be generated by a family of <a href="/wiki/F-seminorm" class="mw-redirect" title="F-seminorm">F-seminorms</a>.<a href="#cite_note-FOOTNOTESwartz199235-65">[53]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"> is some unary <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicate</a> (a true or false statement dependent on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}">) then for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bebbe97b6551af587e1751cca54d9c10bcb09ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.556ex; height:2.509ex;" alt="{\displaystyle z\in X,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+\{x\in X:P(x)\}=\{x\in X:P(x-z)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+\{x\in X:P(x)\}=\{x\in X:P(x-z)\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b7a3a316339e990f584263490687b15da95c13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.196ex; height:2.843ex;" alt="{\displaystyle z+\{x\in X:P(x)\}=\{x\in X:P(x-z)\}.}"><a href="#cite_note-66">[proof 6]</a> So for example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"> denotes "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40c9f7503a2a64b863145627e80dda2098916ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.916ex; height:2.843ex;" alt="{\displaystyle \|x\|<1}">" then for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bebbe97b6551af587e1751cca54d9c10bcb09ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.556ex; height:2.509ex;" alt="{\displaystyle z\in X,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+\{x\in X:\|x\|<1\}=\{x\in X:\|x-z\|<1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+\{x\in X:\|x\|<1\}=\{x\in X:\|x-z\|<1\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3c28ba419ab45b0772dcb89a1b1095b1b10d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.258ex; height:2.843ex;" alt="{\displaystyle z+\{x\in X:\|x\|<1\}=\{x\in X:\|x-z\|<1\}.}"> Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b696e0b457eecf592be8e2fda417810807613c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.351ex; height:2.676ex;" alt="{\displaystyle s\neq 0}"> is a scalar then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\{x\in X:P(x)\}=\left\{x\in X:P\left({\tfrac {1}{s}}x\right)\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mstyle> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\{x\in X:P(x)\}=\left\{x\in X:P\left({\tfrac {1}{s}}x\right)\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8053cbd08d48d87387ac70e020a64a1a5cf87b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.568ex; height:3.343ex;" alt="{\displaystyle s\{x\in X:P(x)\}=\left\{x\in X:P\left({\tfrac {1}{s}}x\right)\right\}.}"> The elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"> of these sets must range over a vector space (that is, over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">) rather than not just a subset or else these equalities are no longer guaranteed; similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> must belong to this vector space (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff47a41328065a0e15e3a7089cfd392ee954e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.909ex; height:2.176ex;" alt="{\displaystyle z\in X}">). <div class="mw-heading mw-heading3"><h3 id="Properties_preserved_by_set_operators">Properties preserved by set operators</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=25" title="Edit section: Properties preserved by set operators">edit</a>]</div> <ul><li>The balanced hull of a compact (respectively, <a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a>, open) set has that same property.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a></li> <li>The <a href="/wiki/Minkowski_sum" class="mw-redirect" title="Minkowski sum">(Minkowski) sum</a> of two compact (respectively, bounded, balanced, convex) sets has that same property.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> But the sum of two closed sets need not be closed.</li> <li>The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need not be closed.<a href="#cite_note-FOOTNOTENariciBeckenstein201167–113-6">[6]</a> And the convex hull of a bounded set need not be bounded.</li></ul> The following table, the color of each cell indicates whether or not a given property of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red. So for instance, since the union of two absorbing sets is again absorbing, the cell in row "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cup S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∪</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cup S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b8d89751b39eee6c6b3f41b90443c409bde84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle R\cup S}">" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in. <table class="wikitable mw-collapsible mw-collapsed"> <caption>Properties preserved by set operators </caption> <tbody><tr> <th rowspan="2">Operation </th> <th colspan="100">Property of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f035e033d7d2c784a07e01448f7605945dfd435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.411ex; height:2.509ex;" alt="{\displaystyle R,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0bcd8516b165aaacb234616d7d2d23478a35be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.146ex; height:2.509ex;" alt="{\displaystyle S,}"> and any other subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that is considered </th></tr> <tr> <th><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing</a> </th> <th><a href="/wiki/Balanced_set" title="Balanced set">Balanced</a> </th> <th><a href="/wiki/Convex_set" title="Convex set">Convex</a> </th> <th><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a> </th> <th>Convex Balanced </th> <th>Vector subspace </th> <th>Open </th> <th>Neighborhood of 0 </th> <th>Closed </th> <th>Closed Balanced </th> <th>Closed Convex </th> <th>Closed Convex Balanced </th> <th><a href="/wiki/Barrelled_set" title="Barrelled set">Barrel</a> </th> <th>Closed Vector subspace </th> <th><a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">Totally bounded</a> </th> <th><a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">Compact</a> </th> <th>Compact Convex </th> <th><a href="/wiki/Relatively_compact" class="mw-redirect" title="Relatively compact">Relatively compact</a> </th> <th><a href="/wiki/Complete_space" class="mw-redirect" title="Complete space">Complete</a> </th> <th><a href="/wiki/Sequentially_complete_space" class="mw-redirect" title="Sequentially complete space">Sequentially Complete</a> </th> <th><a href="/wiki/Banach_disk" class="mw-redirect" title="Banach disk">Banach disk</a> </th> <th><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a> </th> <th><a href="/wiki/Bornivorous_set" title="Bornivorous set">Bornivorous</a> </th> <th><a href="/wiki/Infrabornivorous" class="mw-redirect" title="Infrabornivorous">Infrabornivorous</a> </th> <th><a href="/wiki/Nowhere_dense_set" title="Nowhere dense set">Nowhere dense</a> (in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">) </th> <th><a href="/wiki/Meagre_set" title="Meagre set">Meager</a> </th> <th><a href="/wiki/Separable_space" title="Separable space">Separable</a> </th> <th><a href="/wiki/Metrizable_TVS" class="mw-redirect" title="Metrizable TVS">Pseudometrizable</a> </th> <th>Operation </th></tr> <tr> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cup S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∪</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cup S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b8d89751b39eee6c6b3f41b90443c409bde84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle R\cup S}"> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cup S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∪</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cup S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b8d89751b39eee6c6b3f41b90443c409bde84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle R\cup S}"> </th></tr> <tr> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"> of increasing nonempty <a href="/wiki/Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">chain</a> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∪</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cup }"> of increasing nonempty <a href="/wiki/Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">chain</a> </th></tr> <tr> <th style="text-align:left;">Arbitrary unions (of at least 1 set) </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;">Arbitrary unions (of at least 1 set) </th></tr> <tr> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cap S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∩</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cap S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d2b657f017f3b3fc08ff59bf8f3aaf3c5e28ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle R\cap S}"> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cap S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∩</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cap S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d2b657f017f3b3fc08ff59bf8f3aaf3c5e28ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle R\cap S}"> </th></tr> <tr> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }"> of decreasing nonempty <a href="/wiki/Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">chain</a> </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cap }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }"> of decreasing nonempty <a href="/wiki/Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">chain</a> </th></tr> <tr> <th style="text-align:left;">Arbitrary intersections (of at least 1 set) </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;">Arbitrary intersections (of at least 1 set) </th></tr> <tr> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R+S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c197ee381f4ea9804e5d5b33a0de0997fc99f722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.104ex; height:2.343ex;" alt="{\displaystyle R+S}"> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R+S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c197ee381f4ea9804e5d5b33a0de0997fc99f722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.104ex; height:2.343ex;" alt="{\displaystyle R+S}"> </th></tr> <tr> <th style="text-align:left;">Scalar multiple </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;">Scalar multiple </th></tr> <tr> <th style="text-align:left;">Non-0 scalar multiple </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;">Non-0 scalar multiple </th></tr> <tr> <th style="text-align:left;">Positive scalar multiple </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;">Positive scalar multiple </th></tr> <tr> <th style="text-align:left;"><a href="/wiki/Closure_(topology)" title="Closure (topology)">Closure</a> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><a href="/wiki/Closure_(topology)" title="Closure (topology)">Closure</a> </th></tr> <tr> <th style="text-align:left;"><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a> </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a> </th></tr> <tr> <th style="text-align:left;"><a href="/wiki/Balanced_core" class="mw-redirect" title="Balanced core">Balanced core</a> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><a href="/wiki/Balanced_core" class="mw-redirect" title="Balanced core">Balanced core</a> </th></tr> <tr> <th style="text-align:left;"><a href="/wiki/Balanced_hull" class="mw-redirect" title="Balanced hull">Balanced hull</a> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><a href="/wiki/Balanced_hull" class="mw-redirect" title="Balanced hull">Balanced hull</a> </th></tr> <tr> <th style="text-align:left;"><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a> </th></tr> <tr> <th style="text-align:left;"><a href="/wiki/Convex_balanced_hull" class="mw-redirect" title="Convex balanced hull">Convex balanced hull</a> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><a href="/wiki/Convex_balanced_hull" class="mw-redirect" title="Convex balanced hull">Convex balanced hull</a> </th></tr> <tr> <th style="text-align:left;">Closed balanced hull </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;">Closed balanced hull </th></tr> <tr> <th style="text-align:left;">Closed convex hull </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;">Closed convex hull </th></tr> <tr> <th style="text-align:left;">Closed convex balanced hull </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;">Closed convex balanced hull </th></tr> <tr> <th style="text-align:left;"><a href="/wiki/Linear_span" title="Linear span">Linear span</a> </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <th style="text-align:left;"><a href="/wiki/Linear_span" title="Linear span">Linear span</a> </th></tr> <tr> <th style="text-align:left;">Pre-image under a continuous linear map </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <th style="text-align:left;">Pre-image under a continuous linear map </th></tr> <tr> <th style="text-align:left;">Image under a continuous linear map </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <th style="text-align:left;">Image under a continuous linear map </th></tr> <tr> <th style="text-align:left;">Image under a continuous linear surjection </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <th style="text-align:left;">Image under a continuous linear surjection </th></tr> <tr> <th style="text-align:left;">Non-empty subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="skin-invert table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <td style="background:;"> </td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2 skin-invert"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td> <th style="text-align:left;">Non-empty subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> </th></tr> <tr> <th>Operation </th> <th><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing</a> </th> <th><a href="/wiki/Balanced_set" title="Balanced set">Balanced</a> </th> <th><a href="/wiki/Convex_set" title="Convex set">Convex</a> </th> <th><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a> </th> <th>Convex Balanced </th> <th>Vector subspace </th> <th>Open </th> <th>Neighborhood of 0 </th> <th>Closed </th> <th>Closed Balanced </th> <th>Closed Convex </th> <th>Closed Convex Balanced </th> <th><a href="/wiki/Barrelled_set" title="Barrelled set">Barrel</a> </th> <th>Closed Vector subspace </th> <th><a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">Totally bounded</a> </th> <th><a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">Compact</a> </th> <th>Compact Convex </th> <th><a href="/wiki/Relatively_compact" class="mw-redirect" title="Relatively compact">Relatively compact</a> </th> <th><a href="/wiki/Complete_space" class="mw-redirect" title="Complete space">Complete</a> </th> <th><a href="/wiki/Sequentially_complete_space" class="mw-redirect" title="Sequentially complete space">Sequentially Complete</a> </th> <th><a href="/wiki/Banach_disk" class="mw-redirect" title="Banach disk">Banach disk</a> </th> <th><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a> </th> <th><a href="/wiki/Bornivorous_set" title="Bornivorous set">Bornivorous</a> </th> <th><a href="/wiki/Infrabornivorous" class="mw-redirect" title="Infrabornivorous">Infrabornivorous</a> </th> <th><a href="/wiki/Nowhere_dense_set" title="Nowhere dense set">Nowhere dense</a> (in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">) </th> <th><a href="/wiki/Meagre_set" title="Meagre set">Meager</a> </th> <th><a href="/wiki/Separable_space" title="Separable space">Separable</a> </th> <th><a href="/wiki/Metrizable_TVS" class="mw-redirect" title="Metrizable TVS">Pseudometrizable</a> </th> <th>Operation </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=26" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach space</a> – Normed vector space that is complete</li> <li><a href="/wiki/Complete_field" title="Complete field">Complete field</a> – algebraic structure that is complete relative to a metricPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> – Type of topological vector space</li> <li><a href="/wiki/Liquid_vector_space" class="mw-redirect" title="Liquid vector space">Liquid vector space</a> – Area of mathematics using condensed setsPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed space</a> – Vector space on which a distance is definedPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/Locally_compact_field" title="Locally compact field">Locally compact field</a></li> <li><a href="/wiki/Locally_compact_group" title="Locally compact group">Locally compact group</a> – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Locally_compact_quantum_group" title="Locally compact quantum group">Locally compact quantum group</a> – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a> – A vector space with a topology defined by convex open sets</li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a></li> <li><a href="/wiki/Topological_abelian_group" title="Topological abelian group">Topological abelian group</a> – topological group whose group is abelianPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">Topological field</a> – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/Topological_group" title="Topological group">Topological group</a> – Group that is a topological space with continuous group action</li> <li><a href="/wiki/Topological_module" title="Topological module">Topological module</a></li> <li><a href="/wiki/Topological_ring" title="Topological ring">Topological ring</a> – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Topological_semigroup" title="Topological semigroup">Topological semigroup</a> – semigroup with continuous operationPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=27" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-10"><a href="#cite_ref-10">^</a> The topological properties of course also require that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be a TVS. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is Hausdorff if and only if the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"> is closed (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a <a href="/wiki/T1_space" title="T1 space">T1 space</a>). </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> In fact, this is true for topological group, since the proof does not use the scalar multiplications. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> Also called a metric linear space, which means that it is a real or complex vector space together with a translation-invariant metric for which addition and scalar multiplication are continuous. </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> A series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{\infty }x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{\infty }x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83e196df2a8b33e6fb4bda0c77da38ed193ba12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.87ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{\infty }x_{i}}"> is said to converge in a TVS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> if the sequence of partial sums converges. </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the <a href="/wiki/Particular_point_topology" title="Particular point topology">particular point topology</a> on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+\operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+\operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eff267485c77ceb0bc109ed67a09e204c3915ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.139ex; height:2.843ex;" alt="{\displaystyle S+\operatorname {cl} _{X}\{0\}}"> is compact because it is the image of the compact set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\times \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>×</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\times \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cedd52e8884689b07133d3e0374738545a10931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.139ex; height:2.843ex;" alt="{\displaystyle S\times \operatorname {cl} _{X}\{0\}}"> under the continuous addition map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thickmathspace" /> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a365e2e0209ad7b8c88dc4bffff8532c786d5f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.532ex; height:2.343ex;" alt="{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X.}"> Recall also that the sum of a compact set (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">) and a closed set is closed so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+\operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+\operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eff267485c77ceb0bc109ed67a09e204c3915ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.139ex; height:2.843ex;" alt="{\displaystyle S+\operatorname {cl} _{X}\{0\}}"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d349b099a2e00103b347c5f640a30e0af2a6ee18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{2},}"> the sum of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">-axis and the graph of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\frac {1}{x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {1}{x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fef2b931e4a1f874e96531b5715de189c9e680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.067ex; height:5.176ex;" alt="{\displaystyle y={\frac {1}{x}},}"> which is the complement of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">-axis, is open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066b155c535a38739cc0c4b288324cbb7a4a227a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}.}"> In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}"> the <a href="/wiki/Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} +{\sqrt {2}}\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} +{\sqrt {2}}\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc8889ebac6aa789b95aa5dcf50796b227ced25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.039ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} +{\sqrt {2}}\mathbb {Z} }"> is a countable dense subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> so not closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}"> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Proofs">Proofs</h3>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=28" title="Edit section: Proofs">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-11"><a href="#cite_ref-11">^</a> This condition is satisfied if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"> denotes the set of all topological strings in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b42cadba3d621e09a00a4e133d83ad19f6d548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle (X,\tau ).}"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> This is because every non-empty balanced set must contain the origin and because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \operatorname {Int} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in \operatorname {Int} _{X}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8aed5c43f94081946a8982f566eb4ba256b64f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.559ex; height:2.509ex;" alt="{\displaystyle 0\in \operatorname {Int} _{X}S}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Int} _{X}S=\{0\}\cup \operatorname {Int} _{X}S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <msub> <mi>Int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Int} _{X}S=\{0\}\cup \operatorname {Int} _{X}S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c518a45feec7bd0d694425e5b1bdc03c5bcb6bd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.926ex; height:2.843ex;" alt="{\displaystyle \operatorname {Int} _{X}S=\{0\}\cup \operatorname {Int} _{X}S.}"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> Fix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<r<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<r<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adaa51383bee18b9719d1f44289c12c632aa8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.57ex; height:2.176ex;" alt="{\displaystyle 0<r<1}"> so it remains to show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~rx+(1-r)y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>r</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{0}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~rx+(1-r)y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e470509809a69bd3ca0e5b84e3c300ea47ac77b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.923ex; height:3.509ex;" alt="{\displaystyle w_{0}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~rx+(1-r)y}"> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} _{X}C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{X}C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe565d3edf50e5b957af2da5d5e27f87d66208c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.277ex; height:2.509ex;" alt="{\displaystyle \operatorname {int} _{X}C.}"> By replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C,x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C,x,y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f10bebd25a1b760f527a7f486e8b4bc1290233c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.319ex; height:2.509ex;" alt="{\displaystyle C,x,y}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C-w_{0},x-w_{0},y-w_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>−</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C-w_{0},x-w_{0},y-w_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6515167eb2a4a3f481475a15aeef1b8b8b40f24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.996ex; height:2.509ex;" alt="{\displaystyle C-w_{0},x-w_{0},y-w_{0}}"> if necessary, we may assume without loss of generality that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rx+(1-r)y=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rx+(1-r)y=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9891b60a911b8a17df178c63f9a205c7f48116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.143ex; height:2.843ex;" alt="{\displaystyle rx+(1-r)y=0,}"> and so it remains to show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is a neighborhood of the origin. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\tfrac {r}{r-1}}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>r</mi> <mrow> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\tfrac {r}{r-1}}<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da529cdc47e27baac2ba161bb78dc15cea14c15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.999ex; height:4.009ex;" alt="{\displaystyle s~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\tfrac {r}{r-1}}<0}"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\tfrac {r}{r-1}}x=sx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>r</mi> <mrow> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mi>s</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\tfrac {r}{r-1}}x=sx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af04cec3366331a4d360eb85eb45578ef24b60bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.427ex; height:3.343ex;" alt="{\displaystyle y={\tfrac {r}{r-1}}x=sx.}"> Since scalar multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b696e0b457eecf592be8e2fda417810807613c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.351ex; height:2.676ex;" alt="{\displaystyle s\neq 0}"> is a linear homeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f591cb0a06221871e2529a04bedf5a1a58ac3944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.221ex; height:2.509ex;" alt="{\displaystyle X\to X,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\left({\tfrac {1}{s}}C\right)={\tfrac {1}{s}}\operatorname {cl} _{X}C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mstyle> </mrow> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mstyle> </mrow> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\left({\tfrac {1}{s}}C\right)={\tfrac {1}{s}}\operatorname {cl} _{X}C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5360293d1662202a8526d92c26e382e2d4fdcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.121ex; height:3.343ex;" alt="{\displaystyle \operatorname {cl} _{X}\left({\tfrac {1}{s}}C\right)={\tfrac {1}{s}}\operatorname {cl} _{X}C.}"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \operatorname {int} C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \operatorname {int} C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514491000b9b39962193c17e436565f5d78455cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.168ex; height:2.176ex;" alt="{\displaystyle x\in \operatorname {int} C}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \operatorname {cl} C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>cl</mi> <mo>⁡</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \operatorname {cl} C,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8db911729a5afaf8314183ddfbbd3e3b284f4012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.476ex; height:2.509ex;" alt="{\displaystyle y\in \operatorname {cl} C,}"> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {1}{s}}y\in \operatorname {cl} \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mstyle> </mrow> <mi>y</mi> <mo>∈</mo> <mi>cl</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mstyle> </mrow> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>∩</mo> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {1}{s}}y\in \operatorname {cl} \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ca0aec6d86eeba18d43ffcba93e50a9779efc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.896ex; height:3.343ex;" alt="{\displaystyle x={\tfrac {1}{s}}y\in \operatorname {cl} \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C}"> where because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {int} C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {int} C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ac64f37a2bebc4e52f3fc04b6a06602bde7b7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.997ex; height:2.176ex;" alt="{\displaystyle \operatorname {int} C}"> is open, there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}\in \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mstyle> </mrow> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>∩</mo> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}\in \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021972688f2a039a99ce4eda79f20ef3955ad503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.683ex; height:3.343ex;" alt="{\displaystyle c_{0}\in \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C,}"> which satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sc_{0}\in C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sc_{0}\in C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8277fc206901cd06b0bdebd5326421013c283aba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.405ex; height:2.509ex;" alt="{\displaystyle sc_{0}\in C.}"> Define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20c2288b85fe1e5e807f205314858e908a1ec86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.85ex; height:2.176ex;" alt="{\displaystyle h:X\to X}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto rx+(1-r)sc_{0}=rx-rc_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>r</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>s</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mi>x</mi> <mo>−</mo> <mi>r</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto rx+(1-r)sc_{0}=rx-rc_{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f780a76e047c4d8bcf94f7513fb6f9db4aac60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.249ex; height:2.843ex;" alt="{\displaystyle x\mapsto rx+(1-r)sc_{0}=rx-rc_{0},}"> which is a homeomorphism because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<r<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<r<1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a3ba9d3b53839302868f54ad70ff13f720017f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.217ex; height:2.176ex;" alt="{\displaystyle 0<r<1.}"> The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\left(\operatorname {int} C\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mrow> <mo>(</mo> <mrow> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\left(\operatorname {int} C\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e79156d230edae7c884da7861f5cb8f7652a720e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.533ex; height:2.843ex;" alt="{\displaystyle h\left(\operatorname {int} C\right)}"> is thus an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that moreover contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h(c_{0})=rc_{0}-rc_{0}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle h(c_{0})=rc_{0}-rc_{0}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40944e1b94ae4354c5cc731847499e75571c60e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.275ex; height:2.843ex;" alt="{\textstyle h(c_{0})=rc_{0}-rc_{0}=0.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \operatorname {int} C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \operatorname {int} C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd348f861099346806284ba6f0fe1ba1e4ef880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.845ex; height:2.176ex;" alt="{\displaystyle c\in \operatorname {int} C}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h(c)=rc+(1-r)sc_{0}\in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mi>c</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>s</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle h(c)=rc+(1-r)sc_{0}\in C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb7547bbca469840ca4476e2f8a5abda2b66f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.769ex; height:2.843ex;" alt="{\textstyle h(c)=rc+(1-r)sc_{0}\in C}"> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is convex, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<r<1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<r<1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee29efbc318e8194976c15a7958d9dd12f81471a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.217ex; height:2.509ex;" alt="{\displaystyle 0<r<1,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sc_{0},c\in C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sc_{0},c\in C,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793d4fa259d21c78aeaa88cbea8c1c13d7e7d75b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.446ex; height:2.509ex;" alt="{\displaystyle sc_{0},c\in C,}"> which proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\left(\operatorname {int} C\right)\subseteq C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mrow> <mo>(</mo> <mrow> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>⊆</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\left(\operatorname {int} C\right)\subseteq C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e048cccb47ef6e9c9194651048c5e454758d2105" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.044ex; height:2.843ex;" alt="{\displaystyle h\left(\operatorname {int} C\right)\subseteq C.}"> Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\left(\operatorname {int} C\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mrow> <mo>(</mo> <mrow> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\left(\operatorname {int} C\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e79156d230edae7c884da7861f5cb8f7652a720e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.533ex; height:2.843ex;" alt="{\displaystyle h\left(\operatorname {int} C\right)}"> is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that contains the origin and is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067be67e68f60c53ce83241748d0d6249675c58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}"> Q.E.D. </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> has the trivial topology, so does each of its subsets, which makes them all compact. It is known that a subset of any uniform space is compact if and only if it is complete and totally bounded. </li> <li id="cite_note-ProofSumOfSetAndClosureOf0-51"><a href="#cite_ref-ProofSumOfSetAndClosureOf0_51-0">^</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acce52dffd84d073a24f4606a175da60148fd0c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s+\operatorname {cl} _{X}\{0\}=\operatorname {cl} _{X}(s+\{0\})=\operatorname {cl} _{X}\{s\}\subseteq \operatorname {cl} _{X}S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s+\operatorname {cl} _{X}\{0\}=\operatorname {cl} _{X}(s+\{0\})=\operatorname {cl} _{X}\{s\}\subseteq \operatorname {cl} _{X}S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5aaef45a7fee3f15cdd1994f21202f29f38dbab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.136ex; height:2.843ex;" alt="{\displaystyle s+\operatorname {cl} _{X}\{0\}=\operatorname {cl} _{X}(s+\{0\})=\operatorname {cl} _{X}\{s\}\subseteq \operatorname {cl} _{X}S.}"> Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ddab55c96adeb3691abd632b11873f831900dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.679ex; height:2.843ex;" alt="{\displaystyle S\subseteq S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S,}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is closed then equality holds. Using the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{X}\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{X}\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eb5a3e23f2cd09eaefcc273dabb6e0bf83a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.799ex; height:2.843ex;" alt="{\displaystyle \operatorname {cl} _{X}\{0\}}"> is a vector space, it is readily verified that the complement in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> of any set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> satisfying the equality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f081cd80c08b1eaa80d8be118a6bdc47b038bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.736ex; height:2.843ex;" alt="{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}"> must also satisfy this equality (when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\setminus S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo class="MJX-variant">∖</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\setminus S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b6e9a71fcfbb3ad4209e90c42816daa1766633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.674ex; height:2.843ex;" alt="{\displaystyle X\setminus S}"> is substituted for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">). </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+\{x\in X:P(x)\}=\{z+x:x\in X,P(x)\}=\{z+x:x\in X,P((z+x)-z)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>+</mo> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>+</mo> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+\{x\in X:P(x)\}=\{z+x:x\in X,P(x)\}=\{z+x:x\in X,P((z+x)-z)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6ab674163ff250cbeaabcae2c53d92ea686093" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:78.267ex; height:2.843ex;" alt="{\displaystyle z+\{x\in X:P(x)\}=\{z+x:x\in X,P(x)\}=\{z+x:x\in X,P((z+x)-z)\}}"> and so using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=z+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=z+x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd46d4ad813809f976b5c4d032df5fb519fd4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.512ex; height:2.343ex;" alt="{\displaystyle y=z+x}"> and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z+X=X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mi>X</mi> <mo>=</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+X=X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94939b8635114e89cc9173ccc48f23c7da2d4a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.634ex; height:2.509ex;" alt="{\displaystyle z+X=X,}"> this is equal to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y:y-z\in X,P(y-z)\}=\{y:y\in X,P(y-z)\}=\{y\in X:P(y-z)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>:</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>:</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{y:y-z\in X,P(y-z)\}=\{y:y\in X,P(y-z)\}=\{y\in X:P(y-z)\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871466cc3ceb75e89fd3d921b8b1e8d1adaf0cfe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.782ex; height:2.843ex;" alt="{\displaystyle \{y:y-z\in X,P(y-z)\}=\{y:y\in X,P(y-z)\}=\{y\in X:P(y-z)\}.}"> <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8733090f2d787d03101c3e16dc3f6404f0e7dd4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=29" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTERudin19914-5_§1.3-1"><a href="#cite_ref-FOOTNOTERudin19914-5_§1.3_1-0">^</a> <a 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id="cite_note-FOOTNOTERudin199127_Theorem_1.36-32"><a href="#cite_ref-FOOTNOTERudin199127_Theorem_1.36_32-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 27 Theorem 1.36. </li> <li id="cite_note-FOOTNOTERudin199162-68_§3.8-3.14-33"><a href="#cite_ref-FOOTNOTERudin199162-68_§3.8-3.14_33-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 62-68 §3.8-3.14. </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011177–220-34">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_34-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011177–220_34-1">b</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 177–220. </li> <li id="cite_note-FOOTNOTERudin199138-35">^ <a href="#cite_ref-FOOTNOTERudin199138_35-0">a</a> <a href="#cite_ref-FOOTNOTERudin199138_35-1">b</a> <a href="#cite_ref-FOOTNOTERudin199138_35-2">c</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 38. </li> <li id="cite_note-FOOTNOTESchaeferWolff199935-36"><a href="#cite_ref-FOOTNOTESchaeferWolff199935_36-0">^</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, p. 35. </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011119-120-37"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011119-120_37-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, p. 119-120. </li> <li id="cite_note-FOOTNOTEWilansky201343-38"><a href="#cite_ref-FOOTNOTEWilansky201343_38-0">^</a> <a href="#CITEREFWilansky2013">Wilansky 2013</a>, p. 43. </li> <li id="cite_note-FOOTNOTEWilansky201342-39"><a href="#cite_ref-FOOTNOTEWilansky201342_39-0">^</a> <a href="#CITEREFWilansky2013">Wilansky 2013</a>, p. 42. </li> <li id="cite_note-FOOTNOTERudin199155-40">^ <a href="#cite_ref-FOOTNOTERudin199155_40-0">a</a> <a href="#cite_ref-FOOTNOTERudin199155_40-1">b</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 55. </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011108-41">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011108_41-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011108_41-1">b</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, p. 108. </li> <li id="cite_note-FOOTNOTEJarchow1981101–104-43"><a href="#cite_ref-FOOTNOTEJarchow1981101–104_43-0">^</a> <a href="#CITEREFJarchow1981">Jarchow 1981</a>, pp. 101–104. </li> <li id="cite_note-FOOTNOTESchaeferWolff199938-44">^ <a href="#cite_ref-FOOTNOTESchaeferWolff199938_44-0">a</a> <a href="#cite_ref-FOOTNOTESchaeferWolff199938_44-1">b</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, p. 38. </li> <li id="cite_note-FOOTNOTEConway1990102-45"><a href="#cite_ref-FOOTNOTEConway1990102_45-0">^</a> <a href="#CITEREFConway1990">Conway 1990</a>, p. 102. </li> <li id="cite_note-FOOTNOTENariciBeckenstein201147–66-48">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein201147–66_48-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein201147–66_48-1">b</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein201147–66_48-2">c</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein201147–66_48-3">d</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein201147–66_48-4">e</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein201147–66_48-5">f</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 47–66. </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011156-49"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011156_49-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, p. 156. </li> <li id="cite_note-FOOTNOTESchaeferWolff199912–35-52">^ <a href="#cite_ref-FOOTNOTESchaeferWolff199912–35_52-0">a</a> <a href="#cite_ref-FOOTNOTESchaeferWolff199912–35_52-1">b</a> <a href="#cite_ref-FOOTNOTESchaeferWolff199912–35_52-2">c</a> <a href="#cite_ref-FOOTNOTESchaeferWolff199912–35_52-3">d</a> <a href="#cite_ref-FOOTNOTESchaeferWolff199912–35_52-4">e</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, pp. 12–35. </li> <li id="cite_note-FOOTNOTESchaeferWolff199925-53">^ <a href="#cite_ref-FOOTNOTESchaeferWolff199925_53-0">a</a> <a href="#cite_ref-FOOTNOTESchaeferWolff199925_53-1">b</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, p. 25. </li> <li id="cite_note-FOOTNOTEJarchow198156–73-54">^ <a href="#cite_ref-FOOTNOTEJarchow198156–73_54-0">a</a> <a href="#cite_ref-FOOTNOTEJarchow198156–73_54-1">b</a> <a href="#CITEREFJarchow1981">Jarchow 1981</a>, pp. 56–73. </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011107–112-55"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011107–112_55-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 107–112. </li> <li id="cite_note-FOOTNOTEWilansky201363-56"><a href="#cite_ref-FOOTNOTEWilansky201363_56-0">^</a> <a href="#CITEREFWilansky2013">Wilansky 2013</a>, p. 63. </li> <li id="cite_note-FOOTNOTENariciBeckenstein201119–45-57">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein201119–45_57-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein201119–45_57-1">b</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 19–45. </li> <li id="cite_note-FOOTNOTEWilansky201343–44-59">^ <a href="#cite_ref-FOOTNOTEWilansky201343–44_59-0">a</a> <a href="#cite_ref-FOOTNOTEWilansky201343–44_59-1">b</a> <a href="#cite_ref-FOOTNOTEWilansky201343–44_59-2">c</a> <a href="#CITEREFWilansky2013">Wilansky 2013</a>, pp. 43–44. </li> <li id="cite_note-FOOTNOTENariciBeckenstein201180-60"><a href="#cite_ref-FOOTNOTENariciBeckenstein201180_60-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 80. </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011108–109-61"><a href="#cite_ref-FOOTNOTENariciBeckenstein2011108–109_61-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 108–109. </li> <li id="cite_note-FOOTNOTEJarchow198130–32-62"><a href="#cite_ref-FOOTNOTEJarchow198130–32_62-0">^</a> <a href="#CITEREFJarchow1981">Jarchow 1981</a>, pp. 30–32. </li> <li id="cite_note-FOOTNOTENariciBeckenstein2011109-63">^ <a href="#cite_ref-FOOTNOTENariciBeckenstein2011109_63-0">a</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011109_63-1">b</a> <a href="#cite_ref-FOOTNOTENariciBeckenstein2011109_63-2">c</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, p. 109. </li> <li id="cite_note-FOOTNOTERudin19916-64"><a href="#cite_ref-FOOTNOTERudin19916_64-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 6. </li> <li id="cite_note-FOOTNOTESwartz199235-65"><a href="#cite_ref-FOOTNOTESwartz199235_65-0">^</a> <a href="#CITEREFSwartz1992">Swartz 1992</a>, p. 35. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=30" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdaschErnstKeim1978" class="citation book cs1">Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-08662-8" title="Special:BookSources/978-3-540-08662-8"><bdi>978-3-540-08662-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/297140003">297140003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces%3A+The+Theory+Without+Convexity+Conditions&rft.place=Berlin+New+York&rft.series=Lecture+Notes+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1978&rft_id=info%3Aoclcnum%2F297140003&rft.isbn=978-3-540-08662-8&rft.aulast=Adasch&rft.aufirst=Norbert&rft.au=Ernst%2C+Bruno&rft.au=Keim%2C+Dieter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJarchow1981" class="citation book cs1">Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-519-02224-4" title="Special:BookSources/978-3-519-02224-4"><bdi>978-3-519-02224-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/8210342">8210342</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Locally+convex+spaces&rft.place=Stuttgart&rft.pub=B.G.+Teubner&rft.date=1981&rft_id=info%3Aoclcnum%2F8210342&rft.isbn=978-3-519-02224-4&rft.aulast=Jarchow&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKöthe1983" class="citation book cs1"><a href="/wiki/Gottfried_K%C3%B6the" title="Gottfried Köthe">Köthe, Gottfried</a> (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-64988-2" title="Special:BookSources/978-3-642-64988-2"><bdi>978-3-642-64988-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0248498">0248498</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840293704">840293704</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces+I&rft.place=New+York&rft.series=Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer+Science+%26+Business+Media&rft.date=1983&rft_id=info%3Aoclcnum%2F840293704&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0248498%23id-name%3DMR&rft.isbn=978-3-642-64988-2&rft.aulast=K%C3%B6the&rft.aufirst=Gottfried&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNariciBeckenstein2011" class="citation book cs1">Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1584888666" title="Special:BookSources/978-1584888666"><bdi>978-1584888666</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=Boca+Raton%2C+FL&rft.series=Pure+and+applied+mathematics&rft.edition=Second&rft.pub=CRC+Press&rft.date=2011&rft_id=info%3Aoclcnum%2F144216834&rft.isbn=978-1584888666&rft.aulast=Narici&rft.aufirst=Lawrence&rft.au=Beckenstein%2C+Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1991" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. 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New York, NY: <a href="/wiki/McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-054236-5" title="Special:BookSources/978-0-07-054236-5"><bdi>978-0-07-054236-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.place=New+York%2C+NY&rft.series=International+Series+in+Pure+and+Applied+Mathematics&rft.edition=Second&rft.pub=McGraw-Hill+Science%2FEngineering%2FMath&rft.date=1991&rft_id=info%3Aoclcnum%2F21163277&rft.isbn=978-0-07-054236-5&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="/wiki/Helmut_H._Schaefer" title="Helmut H. 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New York, NY: Springer New York Imprint Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-7155-0" title="Special:BookSources/978-1-4612-7155-0"><bdi>978-1-4612-7155-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=New+York%2C+NY&rft.series=GTM&rft.edition=Second&rft.pub=Springer+New+York+Imprint+Springer&rft.date=1999&rft_id=info%3Aoclcnum%2F840278135&rft.isbn=978-1-4612-7155-0&rft.aulast=Schaefer&rft.aufirst=Helmut+H.&rft.au=Wolff%2C+Manfred+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchechter1996" class="citation book cs1"><a href="/wiki/Eric_Schechter" title="Eric Schechter">Schechter, Eric</a> (1996). 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Dekker. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-8643-4" title="Special:BookSources/978-0-8247-8643-4"><bdi>978-0-8247-8643-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/24909067">24909067</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+Functional+Analysis&rft.place=New+York&rft.pub=M.+Dekker&rft.date=1992&rft_id=info%3Aoclcnum%2F24909067&rft.isbn=978-0-8247-8643-4&rft.aulast=Swartz&rft.aufirst=Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilansky2013" class="citation book cs1"><a href="/wiki/Albert_Wilansky" title="Albert Wilansky">Wilansky, Albert</a> (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49353-4" title="Special:BookSources/978-0-486-49353-4"><bdi>978-0-486-49353-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/849801114">849801114</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Methods+in+Topological+Vector+Spaces&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications%2C+Inc&rft.date=2013&rft_id=info%3Aoclcnum%2F849801114&rft.isbn=978-0-486-49353-4&rft.aulast=Wilansky&rft.aufirst=Albert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=31" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBierstedt1988" class="citation journal cs1"><a href="/w/index.php?title=Klaus-Dieter_Bierstedt&action=edit&redlink=1" class="new" title="Klaus-Dieter Bierstedt (page does not exist)">Bierstedt, Klaus-Dieter</a> (1988). <a rel="nofollow" class="external text" href="https://digital.ub.uni-paderborn.de/hsx/download/pdf/42382?originalFilename=true">"An Introduction to Locally Convex Inductive Limits"</a>. Functional Analysis and Applications. Singapore-New Jersey-Hong Kong: Universitätsbibliothek: 35–133. Retrieved 20 September 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Functional+Analysis+and+Applications&rft.atitle=An+Introduction+to+Locally+Convex+Inductive+Limits&rft.pages=%3Cspan+class%3D%22nowrap%22%3E35-%3C%2Fspan%3E133&rft.date=1988&rft.aulast=Bierstedt&rft.aufirst=Klaus-Dieter&rft_id=https%3A%2F%2Fdigital.ub.uni-paderborn.de%2Fhsx%2Fdownload%2Fpdf%2F42382%3ForiginalFilename%3Dtrue&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1987" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1987) [1981]. Topological Vector Spaces: Chapters 1–5. <a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a>. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-13627-4" title="Special:BookSources/3-540-13627-4"><bdi>3-540-13627-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/17499190">17499190</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces%3A+Chapters+1%E2%80%935&rft.place=Berlin+New+York&rft.series=%C3%89l%C3%A9ments+de+math%C3%A9matique&rft.pub=Springer-Verlag&rft.date=1987&rft_id=info%3Aoclcnum%2F17499190&rft.isbn=3-540-13627-4&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConway1990" class="citation book cs1"><a href="/wiki/John_B._Conway" title="John B. Conway">Conway, John B.</a> (1990). A Course in Functional Analysis. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 96 (2nd ed.). New York: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97245-9" title="Special:BookSources/978-0-387-97245-9"><bdi>978-0-387-97245-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21195908">21195908</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Functional+Analysis&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1990&rft_id=info%3Aoclcnum%2F21195908&rft.isbn=978-0-387-97245-9&rft.aulast=Conway&rft.aufirst=John+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunfordSchwartz1988" class="citation book cs1"><a href="/wiki/Nelson_Dunford" title="Nelson Dunford">Dunford, Nelson</a>; <a href="/wiki/Jacob_T._Schwartz" title="Jacob T. Schwartz">Schwartz, Jacob T.</a> (1988). <a href="/wiki/Linear_Operators_(book)" title="Linear Operators (book)">Linear Operators</a>. Pure and applied mathematics. Vol. 1. New York: <a href="/wiki/Wiley-Interscience" class="mw-redirect" title="Wiley-Interscience">Wiley-Interscience</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-60848-6" title="Special:BookSources/978-0-471-60848-6"><bdi>978-0-471-60848-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/18412261">18412261</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Operators&rft.place=New+York&rft.series=Pure+and+applied+mathematics&rft.pub=Wiley-Interscience&rft.date=1988&rft_id=info%3Aoclcnum%2F18412261&rft.isbn=978-0-471-60848-6&rft.aulast=Dunford&rft.aufirst=Nelson&rft.au=Schwartz%2C+Jacob+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwards1995" class="citation book cs1">Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-68143-6" title="Special:BookSources/978-0-486-68143-6"><bdi>978-0-486-68143-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/30593138">30593138</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis%3A+Theory+and+Applications&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1995&rft_id=info%3Aoclcnum%2F30593138&rft.isbn=978-0-486-68143-6&rft.aulast=Edwards&rft.aufirst=Robert+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrothendieck1973" class="citation book cs1"><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexander</a> (1973). <a rel="nofollow" class="external text" href="https://archive.org/details/topologicalvecto0000grot">Topological Vector Spaces</a>. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-677-30020-7" title="Special:BookSources/978-0-677-30020-7"><bdi>978-0-677-30020-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/886098">886098</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=New+York&rft.pub=Gordon+and+Breach+Science+Publishers&rft.date=1973&rft_id=info%3Aoclcnum%2F886098&rft.isbn=978-0-677-30020-7&rft.aulast=Grothendieck&rft.aufirst=Alexander&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftopologicalvecto0000grot&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHorváth1966" class="citation book cs1"><a href="/wiki/John_Horvath_(mathematician)" title="John Horvath (mathematician)">Horváth, John</a> (1966). Topological Vector Spaces and Distributions. Addison-Wesley series in mathematics. Vol. 1. Reading, MA: Addison-Wesley Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0201029857" title="Special:BookSources/978-0201029857"><bdi>978-0201029857</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces+and+Distributions&rft.place=Reading%2C+MA&rft.series=Addison-Wesley+series+in+mathematics&rft.pub=Addison-Wesley+Publishing+Company&rft.date=1966&rft.isbn=978-0201029857&rft.aulast=Horv%C3%A1th&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKöthe1979" class="citation book cs1"><a href="/wiki/Gottfried_K%C3%B6the" title="Gottfried Köthe">Köthe, Gottfried</a> (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90400-9" title="Special:BookSources/978-0-387-90400-9"><bdi>978-0-387-90400-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/180577972">180577972</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces+II&rft.place=New+York&rft.series=Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer+Science+%26+Business+Media&rft.date=1979&rft_id=info%3Aoclcnum%2F180577972&rft.isbn=978-0-387-90400-9&rft.aulast=K%C3%B6the&rft.aufirst=Gottfried&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1972" class="citation book cs1"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-04166-9" title="Special:BookSources/0-201-04166-9"><bdi>0-201-04166-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+manifolds&rft.place=Reading%2C+Mass.%26ndash%3BLondon%26ndash%3BDon+Mills%2C+Ont.&rft.pub=Addison-Wesley+Publishing+Co.%2C+Inc.&rft.date=1972&rft.isbn=0-201-04166-9&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobertsonRobertson1980" class="citation book cs1">Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. <a href="/w/index.php?title=Cambridge_Tracts_in_Mathematics&action=edit&redlink=1" class="new" title="Cambridge Tracts in Mathematics (page does not exist)">Cambridge Tracts in Mathematics</a>. Vol. 53. Cambridge England: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-29882-7" title="Special:BookSources/978-0-521-29882-7"><bdi>978-0-521-29882-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/589250">589250</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=Cambridge+England&rft.series=Cambridge+Tracts+in+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1980&rft_id=info%3Aoclcnum%2F589250&rft.isbn=978-0-521-29882-7&rft.aulast=Robertson&rft.aufirst=Alex+P.&rft.au=Robertson%2C+Wendy+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="/wiki/Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45352-1" title="Special:BookSources/978-0-486-45352-1"><bdi>978-0-486-45352-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853623322">853623322</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces%2C+Distributions+and+Kernels&rft.place=Mineola%2C+N.Y.&rft.pub=Dover+Publications&rft.date=2006&rft_id=info%3Aoclcnum%2F853623322&rft.isbn=978-0-486-45352-1&rft.aulast=Tr%C3%A8ves&rft.aufirst=Fran%C3%A7ois&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFValdivia1982" class="citation book cs1"><a href="/w/index.php?title=Manuel_Valdivia&action=edit&redlink=1" class="new" title="Manuel Valdivia (page does not exist)">Valdivia, Manuel</a> (1982). Nachbin, Leopoldo (ed.). Topics in Locally Convex Spaces. Vol. 67. Amsterdam New York, N.Y.: <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a> Science Pub. Co. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-087178-3" title="Special:BookSources/978-0-08-087178-3"><bdi>978-0-08-087178-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/316568534">316568534</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+Locally+Convex+Spaces&rft.place=Amsterdam+New+York%2C+N.Y.&rft.pub=Elsevier+Science+Pub.+Co&rft.date=1982&rft_id=info%3Aoclcnum%2F316568534&rft.isbn=978-0-08-087178-3&rft.aulast=Valdivia&rft.aufirst=Manuel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoigt2020" class="citation book cs1"><a href="/w/index.php?title=J%C3%BCrgen_Voigt&action=edit&redlink=1" class="new" title="Jürgen Voigt (page does not exist)">Voigt, Jürgen</a> (2020). A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Cham: <a href="/w/index.php?title=Birkh%C3%A4user_Basel&action=edit&redlink=1" class="new" title="Birkhäuser Basel (page does not exist)">Birkhäuser Basel</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-32945-7" title="Special:BookSources/978-3-030-32945-7"><bdi>978-3-030-32945-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1145563701">1145563701</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+on+Topological+Vector+Spaces&rft.place=Cham&rft.series=Compact+Textbooks+in+Mathematics&rft.pub=Birkh%C3%A4user+Basel&rft.date=2020&rft_id=info%3Aoclcnum%2F1145563701&rft.isbn=978-3-030-32945-7&rft.aulast=Voigt&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+vector+space" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Topological_vector_space&action=edit&section=32" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Topological_vector_spaces" class="extiw" title="commons:Category:Topological vector spaces">Topological vector spaces</a> at Wikimedia Commons</li></ul> <div class="navbox-styles"><style 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a class="mw-selflink selflink">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Topological_vector_spaces_(TVSs)267" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Topological_vector_spaces" title="Template:Topological vector spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Topological_vector_spaces" title="Template talk:Topological vector spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Topological_vector_spaces" title="Special:EditPage/Template:Topological vector spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topological_vector_spaces_(TVSs)267" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Topological vector spaces</a> (TVSs)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach space</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Completeness</a></li> <li><a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Linear functional</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizability</a></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator topologies</a></li> <li><a class="mw-selflink selflink">Topological vector space</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Main results</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph theorem</a></li> <li><a href="/wiki/F._Riesz%27s_theorem" title="F. Riesz's theorem">F. Riesz's</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> (<a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li> <li><a href="/wiki/Vector-valued_Hahn%E2%80%93Banach_theorems" title="Vector-valued Hahn–Banach theorems">Vector-valued Hahn–Banach</a>)</li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Open mapping (Banach–Schauder)</a> <ul><li><a href="/wiki/Bounded_inverse_theorem" class="mw-redirect" title="Bounded inverse theorem">Bounded inverse</a></li></ul></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness (Banach–Steinhaus)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bilinear_operator" class="mw-redirect" title="Bilinear operator">Bilinear operator</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li></ul></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a> <ul><li><a href="/wiki/Almost_open_linear_map" class="mw-redirect" title="Almost open linear map">Almost open</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Closed_linear_operator" title="Closed linear operator">Closed</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Densely_defined_operator" title="Densely defined operator">Densely defined</a></li> <li><a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Discontinuous</a></li></ul></li> <li><a href="/wiki/Topological_homomorphism" title="Topological homomorphism">Topological homomorphism</a></li> <li><a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">Functional</a> <ul><li><a href="/wiki/Linear_form" title="Linear form">Linear</a></li> <li><a href="/wiki/Bilinear_form" title="Bilinear form">Bilinear</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">Sesquilinear</a></li></ul></li> <li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">Norm</a></li> <li><a href="/wiki/Seminorm" title="Seminorm">Seminorm</a></li> <li><a href="/wiki/Sublinear_function" title="Sublinear function">Sublinear function</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex/disk</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing/Radial</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced/Circled</a></li> <li><a href="/wiki/Auxiliary_normed_space" title="Auxiliary normed space">Banach disks</a></li> <li><a href="/wiki/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li> <li><a href="/wiki/Complemented_subspace" title="Complemented subspace">Complemented subspace</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex</a></li> <li><a href="/wiki/Convex_cone" title="Convex cone">Convex cone (subset)</a></li> <li><a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone (subset)</a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Totally_bounded_space#Topological_vector_spaces" title="Totally bounded space">Pre-compact/Totally bounded</a></li> <li><a href="/wiki/Prevalent_and_shy_sets" title="Prevalent and shy sets">Prevalent/Shy</a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial</a></li> <li><a href="/wiki/Star_domain" title="Star domain">Radially convex/Star-shaped</a></li> <li><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set operations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_hull" title="Affine hull">Affine hull</a></li> <li>(<a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li> <li><a href="/wiki/Polar_set" title="Polar set">Polar</a></li> <li>(<a href="/wiki/Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="/wiki/Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of TVSs</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Ptak_space" title="Ptak space">B-complete/Ptak</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li>(<a href="/wiki/Countably_barrelled_space" title="Countably barrelled space">Countably</a>) <a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/BK-space" title="BK-space">BK-space</a></li> <li>(<a href="/wiki/Ultrabornological_space" title="Ultrabornological space">Ultra-</a>) <a href="/wiki/Bornological_space" title="Bornological space">Bornological</a></li> <li><a href="/wiki/Brauner_space" title="Brauner space">Brauner</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Convenient_vector_space" title="Convenient vector space">Convenient</a></li> <li><a href="/wiki/DF-space" title="DF-space">(DF)-space</a></li> <li><a href="/wiki/Distinguished_space" title="Distinguished space">Distinguished</a></li> <li><a href="/wiki/F-space" title="F-space">F-space</a></li> <li><a href="/wiki/FK-AK_space" title="FK-AK space">FK-AK space</a></li> <li><a href="/wiki/FK-space" title="FK-space">FK-space</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a> <ul><li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame Fréchet</a></li></ul></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/Infrabarreled_space" class="mw-redirect" title="Infrabarreled space">Infrabarreled</a></li> <li><a href="/wiki/Interpolation_space" title="Interpolation space">Interpolation space</a></li> <li><a href="/wiki/K-space_(functional_analysis)" title="K-space (functional analysis)">K-space</a></li> <li><a href="/wiki/LB-space" title="LB-space">LB-space</a></li> <li><a href="/wiki/LF-space" title="LF-space">LF-space</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li> <li><a href="/wiki/Mackey_space" title="Mackey space">Mackey</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">(Pseudo)Metrizable</a></li> <li><a href="/wiki/Montel_space" title="Montel space">Montel</a></li> <li><a href="/wiki/Quasibarrelled_space" class="mw-redirect" title="Quasibarrelled space">Quasibarrelled</a></li> <li><a href="/wiki/Quasi-complete" class="mw-redirect" title="Quasi-complete">Quasi-complete</a></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a></li> <li><a href="/wiki/Semi-reflexive_space" title="Semi-reflexive space">Semi-</a>) <a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz</a></li> <li><a href="/wiki/Schwartz_TVS" class="mw-redirect" title="Schwartz TVS">Schwartz</a></li> <li><a href="/wiki/Semi-complete" class="mw-redirect" title="Semi-complete">Semi-complete</a></li> <li><a href="/wiki/Smith_space" title="Smith space">Smith</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li> <li>(<a href="/wiki/B-convex_space" title="B-convex space">B</a></li> <li><a href="/wiki/Strictly_convex_space" title="Strictly convex space">Strictly</a></li> <li><a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li> <li>(<a href="/wiki/Quasi-ultrabarrelled_space" title="Quasi-ultrabarrelled space">Quasi-</a>) <a href="/wiki/Ultrabarrelled_space" title="Ultrabarrelled space">Ultrabarrelled</a></li> <li><a href="/wiki/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li><a href="/wiki/Webbed_space" title="Webbed space">Webbed</a></li> <li><a href="/wiki/Approximation_property" title="Approximation property">With the approximation property</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Topological_vector_spaces" title="Category:Topological vector spaces">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" 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François Topological vector spaces, distributions and kernels\"] = 1,\n [\"Valdivia Topics in Locally Convex Spaces\"] = 1,\n [\"Visible anchor\"] = 20,\n [\"Voigt A Course on Topological Vector Spaces\"] = 1,\n [\"Wilansky Modern Methods in Topological Vector Spaces\"] = 1,\n [\"Ya\"] = 358,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-b766959bd-r6hmk","timestamp":"20250215042832","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Topological vector space","url":"https:\/\/en.wikipedia.org\/wiki\/Topological_vector_space","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1455249","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1455249","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-03-24T07:46:08Z","dateModified":"2024-10-04T12:47:12Z","headline":"vector space equipped with a compatible topology"}</script> </body> </html>