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[o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition_and_basic_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_and_basic_properties"> <div class="vector-toc-text"> 1 Definition and basic properties </div> </a> <ul id="toc-Definition_and_basic_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bases,_vector_coordinates,_and_subspaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bases,_vector_coordinates,_and_subspaces"> <div class="vector-toc-text"> 2 Bases, vector coordinates, and subspaces </div> </a> <ul id="toc-Bases,_vector_coordinates,_and_subspaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 3 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </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-Arrows_in_the_plane" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arrows_in_the_plane"> <div class="vector-toc-text"> 4.1 Arrows in the plane </div> </a> <ul id="toc-Arrows_in_the_plane-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordered_pairs_of_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordered_pairs_of_numbers"> <div class="vector-toc-text"> 4.2 Ordered pairs of numbers </div> </a> <ul id="toc-Ordered_pairs_of_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinate_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coordinate_space"> <div class="vector-toc-text"> 4.3 Coordinate space </div> </a> <ul id="toc-Coordinate_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_numbers_and_other_field_extensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_numbers_and_other_field_extensions"> <div class="vector-toc-text"> 4.4 Complex numbers and other field extensions </div> </a> <ul id="toc-Complex_numbers_and_other_field_extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Function_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Function_spaces"> <div class="vector-toc-text"> 4.5 Function spaces </div> </a> <ul id="toc-Function_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_equations"> <div class="vector-toc-text"> 4.6 Linear equations </div> </a> <ul id="toc-Linear_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Linear_maps_and_matrices" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_maps_and_matrices"> <div class="vector-toc-text"> 5 Linear maps and matrices </div> </a> <button aria-controls="toc-Linear_maps_and_matrices-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Linear maps and matrices subsection </button> <ul id="toc-Linear_maps_and_matrices-sublist" class="vector-toc-list"> <li id="toc-Matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrices"> <div class="vector-toc-text"> 5.1 Matrices </div> </a> <ul id="toc-Matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenvalues_and_eigenvectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenvalues_and_eigenvectors"> <div class="vector-toc-text"> 5.2 Eigenvalues and eigenvectors </div> </a> <ul id="toc-Eigenvalues_and_eigenvectors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_constructions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_constructions"> <div class="vector-toc-text"> 6 Basic constructions </div> </a> <button aria-controls="toc-Basic_constructions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Basic constructions subsection </button> <ul id="toc-Basic_constructions-sublist" class="vector-toc-list"> <li id="toc-Subspaces_and_quotient_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subspaces_and_quotient_spaces"> <div class="vector-toc-text"> 6.1 Subspaces and quotient spaces </div> </a> <ul id="toc-Subspaces_and_quotient_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Direct_product_and_direct_sum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Direct_product_and_direct_sum"> <div class="vector-toc-text"> 6.2 Direct product and direct sum </div> </a> <ul id="toc-Direct_product_and_direct_sum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_product"> <div class="vector-toc-text"> 6.3 Tensor product </div> </a> <ul id="toc-Tensor_product-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vector_spaces_with_additional_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vector_spaces_with_additional_structure"> <div class="vector-toc-text"> 7 Vector spaces with additional structure </div> </a> <button aria-controls="toc-Vector_spaces_with_additional_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Vector spaces with additional structure subsection </button> <ul id="toc-Vector_spaces_with_additional_structure-sublist" class="vector-toc-list"> <li id="toc-Normed_vector_spaces_and_inner_product_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normed_vector_spaces_and_inner_product_spaces"> <div class="vector-toc-text"> 7.1 Normed vector spaces and inner product spaces </div> </a> <ul id="toc-Normed_vector_spaces_and_inner_product_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_vector_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_vector_spaces"> <div class="vector-toc-text"> 7.2 Topological vector spaces </div> </a> <ul id="toc-Topological_vector_spaces-sublist" class="vector-toc-list"> <li id="toc-Banach_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Banach_spaces"> <div class="vector-toc-text"> 7.2.1 Banach spaces </div> </a> <ul id="toc-Banach_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hilbert_spaces"> <div class="vector-toc-text"> 7.2.2 Hilbert spaces </div> </a> <ul id="toc-Hilbert_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algebras_over_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebras_over_fields"> <div class="vector-toc-text"> 7.3 Algebras over fields </div> </a> <ul id="toc-Algebras_over_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_structures" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_structures"> <div class="vector-toc-text"> 8 Related structures </div> </a> <button aria-controls="toc-Related_structures-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Related structures subsection </button> <ul id="toc-Related_structures-sublist" class="vector-toc-list"> <li id="toc-Vector_bundles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_bundles"> <div class="vector-toc-text"> 8.1 Vector bundles </div> </a> <ul id="toc-Vector_bundles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modules"> <div class="vector-toc-text"> 8.2 Modules </div> </a> <ul id="toc-Modules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_and_projective_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_and_projective_spaces"> <div class="vector-toc-text"> 8.3 Affine and projective spaces </div> </a> <ul id="toc-Affine_and_projective_spaces-sublist" class="vector-toc-list"> </ul> </li> </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"> 9 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </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"> 10 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 11 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebra"> <div class="vector-toc-text"> 11.1 Algebra </div> </a> <ul id="toc-Algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analysis"> <div class="vector-toc-text"> 11.2 Analysis </div> </a> <ul id="toc-Analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_references"> <div class="vector-toc-text"> 11.3 Historical references </div> </a> <ul id="toc-Historical_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_references"> <div class="vector-toc-text"> 11.4 Further references </div> </a> <ul id="toc-Further_references-sublist" class="vector-toc-list"> </ul> </li> </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"> 12 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">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 78 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-78" aria-hidden="true" > 78 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vektorruimte" title="Vektorruimte – Afrikaans" lang="af" hreflang="af" data-title="Vektorruimte" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><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" title="فضاء متجهي – Arabic" lang="ar" hreflang="ar" data-title="فضاء متجهي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_vectorial" title="Espaciu vectorial – Asturian" lang="ast" hreflang="ast" data-title="Espaciu vectorial" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%A6%E0%A6%BF%E0%A6%95_%E0%A6%B0%E0%A6%BE%E0%A6%B6%E0%A6%BF%E0%A6%B0_%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="সদিক রাশির বীজগণিত – Bangla" lang="bn" hreflang="bn" data-title="সদিক রাশির বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hi%C3%B2ng-li%C5%8Dng_khong-kan" title="Hiòng-liōng khong-kan – Minnan" lang="nan" hreflang="nan" data-title="Hiòng-liōng khong-kan" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%D1%8B_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%A1" title="Векторлы арауыҡ – Bashkir" lang="ba" hreflang="ba" data-title="Векторлы арауыҡ" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B0%D1%81%D1%82%D0%BE%D1%80%D0%B0" title="Вектарная прастора – Belarusian" lang="be" hreflang="be" data-title="Вектарная прастора" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%9F%E0%A4%B0_%E0%A4%B8%E0%A5%8D%E0%A4%AA%E0%A5%87%E0%A4%B8" title="वेक्टर स्पेस – Bhojpuri" lang="bh" hreflang="bh" data-title="वेक्टर स्पेस" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE_%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="Линейно пространство – Bulgarian" lang="bg" hreflang="bg" data-title="Линейно пространство" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor – Bosnian" lang="bs" hreflang="bs" data-title="Vektorski prostor" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ca.wikipedia.org/wiki/Espai_vectorial" title="Espai vectorial – Catalan" lang="ca" hreflang="ca" data-title="Espai vectorial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%D0%B0_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Векторла уçлăх – Chuvash" lang="cv" hreflang="cv" data-title="Векторла уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vektorov%C3%BD_prostor" title="Vektorový prostor – Czech" lang="cs" hreflang="cs" data-title="Vektorový prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_fector" title="Gofod fector – Welsh" lang="cy" hreflang="cy" data-title="Gofod fector" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Vektorrum" title="Vektorrum – Danish" lang="da" hreflang="da" data-title="Vektorrum" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Vektorraum" title="Vektorraum – German" lang="de" hreflang="de" data-title="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/Vektorruum" title="Vektorruum – Estonian" lang="et" hreflang="et" data-title="Vektorruum" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CE%BD%CF%85%CF%83%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Διανυσματικός χώρος – Greek" lang="el" hreflang="el" data-title="Διανυσματικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_vectorial" title="Espacio vectorial – Spanish" lang="es" hreflang="es" data-title="Espacio vectorial" 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/Vektora_spaco" title="Vektora spaco – Esperanto" lang="eo" hreflang="eo" data-title="Vektora spaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bektore_espazio" title="Bektore espazio – Basque" lang="eu" hreflang="eu" data-title="Bektore espazio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</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" title="فضای برداری – Persian" lang="fa" hreflang="fa" data-title="فضای برداری" 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" title="Espace vectoriel – French" lang="fr" hreflang="fr" data-title="Espace vectoriel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sp%C3%A1s_veicteoireach" title="Spás veicteoireach – Irish" lang="ga" hreflang="ga" data-title="Spás veicteoireach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_vectorial" title="Espazo vectorial – Galician" lang="gl" hreflang="gl" data-title="Espazo vectorial" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간 – Korean" lang="ko" hreflang="ko" data-title="벡터 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8E%D5%A5%D5%AF%D5%BF%D5%B8%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Վեկտորական տարածություն – Armenian" lang="hy" hreflang="hy" data-title="Վեկտորական տարածություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6_%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="सदिश बीजगणित – Hindi" lang="hi" hreflang="hi" data-title="सदिश बीजगणित" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor – Croatian" lang="hr" hreflang="hr" data-title="Vektorski prostor" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Vektorala_spaco" title="Vektorala spaco – Ido" lang="io" hreflang="io" data-title="Vektorala spaco" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_vektor" title="Ruang vektor – Indonesian" lang="id" hreflang="id" data-title="Ruang vektor" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Spatio_vectorial" title="Spatio vectorial – Interlingua" lang="ia" hreflang="ia" data-title="Spatio vectorial" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vigurr%C3%BAm" title="Vigurrúm – Icelandic" lang="is" hreflang="is" data-title="Vigurrúm" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_vettoriale" title="Spazio vettoriale – Italian" lang="it" hreflang="it" data-title="Spazio vettoriale" 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" title="מרחב וקטורי – Hebrew" lang="he" hreflang="he" data-title="מרחב וקטורי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B4%D1%83%D0%BA_%D0%BC%D0%B5%D0%B9%D0%BA%D0%B8%D0%BD%D0%B4%D0%B8%D0%BA" title="Вектордук мейкиндик – Kyrgyz" lang="ky" hreflang="ky" data-title="Вектордук мейкиндик" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%80%E0%BA%A7%E0%BA%B1%E0%BA%81%E0%BB%80%E0%BA%95%E0%BA%B5" title="ເວັກເຕີ – Lao" lang="lo" hreflang="lo" data-title="ເວັກເຕີ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Spatium_vectoriale" title="Spatium vectoriale – Latin" lang="la" hreflang="la" data-title="Spatium vectoriale" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vektoru_telpa" title="Vektoru telpa – Latvian" lang="lv" hreflang="lv" data-title="Vektoru telpa" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vektorin%C4%97_erdv%C4%97" title="Vektorinė erdvė – Lithuanian" lang="lt" hreflang="lt" data-title="Vektorinė erdvė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Spazzi_vettorial" title="Spazzi vettorial – Lombard" lang="lmo" hreflang="lmo" data-title="Spazzi vettorial" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Vektort%C3%A9r" title="Vektortér – Hungarian" lang="hu" hreflang="hu" data-title="Vektortér" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор – Macedonian" lang="mk" hreflang="mk" data-title="Векторски простор" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%A6%E0%B4%BF%E0%B4%B6%E0%B4%B8%E0%B4%AE%E0%B4%B7%E0%B5%8D%E0%B4%9F%E0%B4%BF" title="സദിശസമഷ്ടി – Malayalam" lang="ml" hreflang="ml" data-title="സദിശസമഷ്ടി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_vektor" title="Ruang vektor – Malay" lang="ms" hreflang="ms" data-title="Ruang vektor" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vectorruimte" title="Vectorruimte – Dutch" lang="nl" hreflang="nl" data-title="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/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E7%A9%BA%E9%96%93" title="ベクトル空間 – Japanese" lang="ja" hreflang="ja" data-title="ベクトル空間" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vektorrom" title="Vektorrom – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Vektorrom" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Vektorrom" title="Vektorrom – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Vektorrom" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Espaci_vectoriau" title="Espaci vectoriau – Occitan" lang="oc" hreflang="oc" data-title="Espaci vectoriau" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A9%88%E0%A8%95%E0%A8%9F%E0%A8%B0_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਵੈਕਟਰ ਸਪੇਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਵੈਕਟਰ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%88%DB%8C%DA%A9%D9%B9%D8%B1_%D8%B3%D9%BE%DB%8C%D8%B3" title="ویکٹر سپیس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ویکٹر سپیس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Spassi_vetorial" title="Spassi vetorial – Piedmontese" lang="pms" hreflang="pms" data-title="Spassi vetorial" 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_liniowa" title="Przestrzeń liniowa – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń liniowa" 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_vetorial" title="Espaço vetorial – Portuguese" lang="pt" hreflang="pt" data-title="Espaço vetorial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_vectorial" title="Spațiu vectorial – Romanian" lang="ro" hreflang="ro" data-title="Spațiu vectorial" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%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="Векторное пространство – Russian" lang="ru" hreflang="ru" data-title="Векторное пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hap%C3%ABsira_vektoriale" title="Hapësira vektoriale – Albanian" lang="sq" hreflang="sq" data-title="Hapësira vektoriale" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Spazziu_vitturiali" title="Spazziu vitturiali – Sicilian" lang="scn" hreflang="scn" data-title="Spazziu vitturiali" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vector_space" title="Vector space – Simple English" lang="en-simple" hreflang="en-simple" data-title="Vector space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vektorov%C3%BD_priestor" title="Vektorový priestor – Slovak" lang="sk" hreflang="sk" data-title="Vektorový priestor" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor – Slovenian" lang="sl" hreflang="sl" data-title="Vektorski prostor" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%D8%A6%D8%A7%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D8%A8%DA%95%DB%95%DA%A9%D8%A7%D9%86" title="بۆشاییی ئاڕاستەبڕەکان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بۆشاییی ئاڕاستەبڕەکان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Векторски простор – Serbian" lang="sr" hreflang="sr" data-title="Векторски простор" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Vektorski_prostor" title="Vektorski prostor – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Vektorski prostor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vektoriavaruus" title="Vektoriavaruus – Finnish" lang="fi" hreflang="fi" data-title="Vektoriavaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rt_rum" title="Linjärt rum – Swedish" lang="sv" hreflang="sv" data-title="Linjärt rum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espasyong_bektor" title="Espasyong bektor – Tagalog" lang="tl" hreflang="tl" data-title="Espasyong bektor" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%A9%E0%AF%8D_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF" title="திசையன் வெளி – Tamil" lang="ta" hreflang="ta" data-title="திசையன் வெளி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Vekt%C3%B6r_uzay%C4%B1" title="Vektör uzayı – Turkish" lang="tr" hreflang="tr" data-title="Vektör uzayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%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="Векторний простір – Ukrainian" lang="uk" hreflang="uk" data-title="Векторний простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B3%D9%85%D8%AA%DB%8C%DB%81_%D9%85%DA%A9%D8%A7%DA%BA" title="سمتیہ مکاں – Urdu" lang="ur" hreflang="ur" data-title="سمتیہ مکاں" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Spasio_vetorial" title="Spasio vetorial – Venetian" lang="vec" hreflang="vec" data-title="Spasio vetorial" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_vect%C6%A1" title="Không gian vectơ – Vietnamese" lang="vi" hreflang="vi" data-title="Không gian vectơ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%9F%A2%E9%87%8F%E7%A9%BA%E9%96%93" title="矢量空間 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="矢量空間" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间 – Wu" lang="wuu" hreflang="wuu" data-title="向量空间" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="向量空間 – Cantonese" lang="yue" hreflang="yue" data-title="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/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间 – Chinese" lang="zh" hreflang="zh" data-title="向量空间" 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/Q125977#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/Vector_space" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Vector_space" rel="discussion" title="Discuss improvements to the content 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Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></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">Algebraic structure in linear algebra</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">Not to be confused with <a href="/wiki/Vector_field" title="Vector field">Vector field</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Linear space" redirects here. For a structure in incidence geometry, see <a href="/wiki/Linear_space_(geometry)" title="Linear space (geometry)">Linear space (geometry)</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vector_add_scale.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Vector_add_scale.svg/200px-Vector_add_scale.svg.png" decoding="async" width="200" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Vector_add_scale.svg/300px-Vector_add_scale.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Vector_add_scale.svg/400px-Vector_add_scale.svg.png 2x" data-file-width="530" data-file-height="294" /></a><figcaption>Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Physics" title="Physics">physics</a>, a vector space (also called a linear space) is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> whose elements, often called <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>, can be added together and multiplied ("scaled") by numbers called <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>. The operations of vector addition and <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> must satisfy certain requirements, called vector <a href="/wiki/Axiom" title="Axiom">axioms</a>. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> and <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>. Scalars can also be, more generally, elements of any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. Vector spaces generalize <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vectors</a>, which allow modeling of <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantities</a> (such as <a href="/wiki/Force" title="Force">forces</a> and <a href="/wiki/Velocity" title="Velocity">velocity</a>) that have not only a <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>, but also a <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">direction</a>. The concept of vector spaces is fundamental for <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, together with the concept of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying <a href="/wiki/Systems_of_linear_equations" class="mw-redirect" title="Systems of linear equations">systems of linear equations</a>. Vector spaces are characterized by their <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a>, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a>). A vector space is finite-dimensional if its dimension is a <a href="/wiki/Natural_number" title="Natural number">natural number</a>. Otherwise, it is infinite-dimensional, and its dimension is an <a href="/wiki/Transfinite_number" title="Transfinite number">infinite cardinal</a>. Finite-dimensional vector spaces occur naturally in <a href="/wiki/Geometry" title="Geometry">geometry</a> and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> are <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably</a> infinite-dimensional vector spaces, and many <a href="/wiki/Function_space" title="Function space">function spaces</a> have the <a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a> as a dimension. Many vector spaces that are considered in mathematics are also endowed with other <a href="/wiki/Mathematical_structure" title="Mathematical structure">structures</a>. This is the case of <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebras</a>, which include <a href="/wiki/Field_extension" title="Field extension">field extensions</a>, polynomial rings, <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebras</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a>. 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id="Definition_and_basic_properties">Definition and basic properties</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=1" title="Edit section: Definition and basic properties">edit</a>]</div> In this article, vectors are represented in boldface to distinguish them from scalars.<a href="#cite_note-1">[nb 1]</a><a href="#cite_note-FOOTNOTELang2002-2">[1]</a> A vector space over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> F is a non-empty <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> V together with a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> and a <a href="/wiki/Binary_function" title="Binary function">binary function</a> that satisfy the eight <a href="/wiki/Axiom" title="Axiom">axioms</a> listed below. In this context, the elements of V are commonly called vectors, and the elements of F are called scalars.<a href="#cite_note-FOOTNOTEBrown199186-3">[2]</a> <ul><li>The binary operation, called vector addition or simply addition assigns to any two vectors v and w in V a third vector in V which is commonly written as v + w, and called the sum of these two vectors.</li></ul> <ul><li>The binary function, called <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>, assigns to any scalar a in F and any vector v in V another vector in V, which is denoted av.<a href="#cite_note-4">[nb 2]</a></li></ul> To have a vector space, the eight following <a href="/wiki/Axiom" title="Axiom">axioms</a> must be satisfied for every u, v and w in V, and a and b in F.<a href="#cite_note-FOOTNOTERoman2005ch._1,_p._27-5">[3]</a> <table border="0" class="wikitable" style="max-width:50em"> <tbody><tr> <th>Axiom </th> <th>Statement </th></tr> <tr> <td><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a> of vector addition</td> <td>u + (v + w) = (u + v) + w </td></tr> <tr style="background:#F8F4FF;"> <td><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a> of vector addition</td> <td>u + v = v + u </td></tr> <tr> <td><a href="/wiki/Identity_element" title="Identity element">Identity element</a> of vector addition</td> <td>There exists an element 0 ∈ V, called the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a>, such that v + 0 = v for all v ∈ V. </td></tr> <tr style="background:#F8F4FF;"> <td><a href="/wiki/Inverse_element" title="Inverse element">Inverse elements</a> of vector addition</td> <td>For every v ∈ V, there exists an element −v ∈ V, called the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> of v, such that v + (−v) = 0. </td></tr> <tr> <td>Compatibility of scalar multiplication with field multiplication</td> <td>a(bv) = (ab)v <a href="#cite_note-6">[nb 3]</a> </td></tr> <tr style="background:#F8F4FF;"> <td>Identity element of scalar multiplication</td> <td>1v = v, where 1 denotes the <a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> in F. </td></tr> <tr> <td><a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a> of scalar multiplication with respect to vector addition  </td> <td>a(u + v) = au + av </td></tr> <tr style="background:#F8F4FF;"> <td>Distributivity of scalar multiplication with respect to field addition</td> <td>(a + b)v = av + bv </td></tr></tbody></table> When the scalar field is the <a href="/wiki/Real_number" title="Real number">real numbers</a>, the vector space is called a real vector space, and when the scalar field is the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, the vector space is called a complex vector space.<a href="#cite_note-FOOTNOTEBrown199187-7">[4]</a> These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered. Such a vector space is called an F-vector space or a vector space over F.<a href="#cite_note-FOOTNOTESpringer2000[httpsbooksgooglecombooksidCes-AAAAQBAJpgPA185_185]Brown199186-8">[5]</a> An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from the field F into the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of this group.<a href="#cite_note-FOOTNOTEAtiyahMacdonald196917-9">[6]</a> Subtraction of two vectors can be defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} -\mathbf {w} =\mathbf {v} +(-\mathbf {w} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} -\mathbf {w} =\mathbf {v} +(-\mathbf {w} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d7e4e2d1ff332517fe5afbda913e8f33df1ada" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.728ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} -\mathbf {w} =\mathbf {v} +(-\mathbf {w} ).}"> Direct consequences of the axioms include that, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6fe9319682deb1cd7944bfec2a95227c49045ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle s\in F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \in V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \in V,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794f60364056e7f3afbf6dbd23f17740438e1ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.686ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} \in V,}"> one has <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\mathbf {v} =\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\mathbf {v} =\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba3803fd77a3adfb4d6afeaabfd65d210bc0daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.655ex; height:2.509ex;" alt="{\displaystyle 0\mathbf {v} =\mathbf {0} ,}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\mathbf {0} =\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\mathbf {0} =\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39449d1c7d13405130f3d34c1cfa1895f5dfbdcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.509ex; height:2.509ex;" alt="{\displaystyle s\mathbf {0} =\mathbf {0} ,}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)\mathbf {v} =-\mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)\mathbf {v} =-\mathbf {v} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb25fcc46e503f132a6a6f2717f3b00b2bb9939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.155ex; height:2.843ex;" alt="{\displaystyle (-1)\mathbf {v} =-\mathbf {v} ,}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\mathbf {v} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\mathbf {v} =\mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87454385b059c0b19b950699c785eb3013514270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.937ex; height:2.176ex;" alt="{\displaystyle s\mathbf {v} =\mathbf {0} }"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=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=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7903b8069a44c70f6f96511675bdd9a4ff200ed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=0}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =\mathbf {0} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\mathbf {0} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78b3b71ce5ff4fed3781de611bd966408b90bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.493ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} =\mathbf {0} .}"></li></ul> Even more concisely, a vector space is a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>.<a href="#cite_note-FOOTNOTEBourbaki1998§1.1,_Definition_2-10">[7]</a> <div class="mw-heading mw-heading2"><h2 id="Bases,_vector_coordinates,_and_subspaces">Bases, vector coordinates, and subspaces</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=2" title="Edit section: Bases, vector coordinates, and subspaces">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vector_components_and_base_change.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Vector_components_and_base_change.svg/200px-Vector_components_and_base_change.svg.png" decoding="async" width="200" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Vector_components_and_base_change.svg/300px-Vector_components_and_base_change.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Vector_components_and_base_change.svg/400px-Vector_components_and_base_change.svg.png 2x" data-file-width="244" data-file-height="240" /></a><figcaption>A vector v in R2 (blue) expressed in terms of different bases: using the <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> of R2: v = xe1 + ye2 (black), and using a different, non-<a href="/wiki/Orthogonal_vector" class="mw-redirect" title="Orthogonal vector">orthogonal</a> basis: v = f1 + f2 (red).</figcaption></figure> <dl><dt><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></dt> <dd>Given a set G of elements of a F-vector space V, a linear combination of elements of G is an element of V of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\mathbf {g} _{1}+a_{2}\mathbf {g} _{2}+\cdots +a_{k}\mathbf {g} _{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\mathbf {g} _{1}+a_{2}\mathbf {g} _{2}+\cdots +a_{k}\mathbf {g} _{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4875d560ecf3338a56c7bf33c46cf1f80edec4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.985ex; height:2.509ex;" alt="{\displaystyle a_{1}\mathbf {g} _{1}+a_{2}\mathbf {g} _{2}+\cdots +a_{k}\mathbf {g} _{k},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{k}\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{k}\in F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f66f5987b771668981dd85c7ada3ce3b3bea551" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.362ex; height:2.509ex;" alt="{\displaystyle a_{1},\ldots ,a_{k}\in F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} _{1},\ldots ,\mathbf {g} _{k}\in G.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>G</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{1},\ldots ,\mathbf {g} _{k}\in G.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0817f411cb5afaaba86961d8b13e31b0a66d68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.309ex; height:2.676ex;" alt="{\displaystyle \mathbf {g} _{1},\ldots ,\mathbf {g} _{k}\in G.}"> The scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9c0425acf7d429b00d87e9a650d8f277dfe7b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.781ex; height:2.009ex;" alt="{\displaystyle a_{1},\ldots ,a_{k}}"> are called the coefficients of the linear combination.<a href="#cite_note-FOOTNOTEBrown199194-11">[8]</a></dd> <dt><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></dt> <dd>The elements of a subset G of a F-vector space V are said to be linearly independent if no element of G can be written as a linear combination of the other elements of G. Equivalently, they are linearly independent if two linear combinations of elements of G define the same element of V if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero.<a href="#cite_note-FOOTNOTEBrown199199–101-12">[9]</a></dd> <dt><a href="/wiki/Linear_subspace" title="Linear subspace">Linear subspace</a></dt> <dd>A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of W by a scalar belong to W.<a href="#cite_note-FOOTNOTEBrown199192-13">[10]</a> This implies that every linear combination of elements of W belongs to W. A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied.<a href="#cite_note-FOOTNOTEStollWong1968[httpsbooksgooglecombooksidgLbiBQAAQBAJpgPA14_14]-14">[11]</a> The closure property also implies that every <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of linear subspaces is a linear subspace.<a href="#cite_note-FOOTNOTEStollWong1968[httpsbooksgooglecombooksidgLbiBQAAQBAJpgPA14_14]-14">[11]</a></dd> <dt><a href="/wiki/Linear_span" title="Linear span">Linear span</a></dt> <dd>Given a subset G of a vector space V, the linear span or simply the span of G is the smallest linear subspace of V that contains G, in the sense that it is the intersection of all linear subspaces that contain G. The span of G is also the set of all linear combinations of elements of G. If W is the span of G, one says that G spans or generates W, and that G is a <a href="/wiki/Spanning_set" class="mw-redirect" title="Spanning set">spanning set</a> or a generating set of W.<a href="#cite_note-FOOTNOTERoman200541–42-15">[12]</a></dd> <dt><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a> and <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a></dt> <dd>A subset of a vector space is a basis if its elements are linearly independent and span the vector space.<a href="#cite_note-FOOTNOTELang198710–11AntonRorres2010[httpsbooksgooglecombooksid1PJ-WHepeBsCpgPA212_212]-16">[13]</a> Every vector space has at least one basis, or many in general (see <a href="/wiki/Basis_(linear_algebra)#Proof_that_every_vector_space_has_a_basis" title="Basis (linear algebra)">Basis (linear algebra) § Proof that every vector space has a basis</a>).<a href="#cite_note-FOOTNOTEBlass1984-17">[14]</a> Moreover, all bases of a vector space have the same <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>, which is called the dimension of the vector space (see <a href="/wiki/Dimension_theorem_for_vector_spaces" title="Dimension theorem for vector spaces">Dimension theorem for vector spaces</a>).<a href="#cite_note-FOOTNOTEJoshi1989[httpsbooksgooglecombooksidRM1D3mFw2u0CpgPA450_450]-18">[15]</a> This is a fundamental property of vector spaces, which is detailed in the remainder of the section.</dd></dl> Bases are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called <a href="/wiki/Hamel_bases" class="mw-redirect" title="Hamel bases">Hamel bases</a>, depends on the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. It follows that, in general, no base can be explicitly described.<a href="#cite_note-FOOTNOTEHeil2011[httpsbooksgooglecombooksidprfuUT0Sw-ACpgPA126_126]-19">[16]</a> For example, the <a href="/wiki/Real_number" title="Real number">real numbers</a> form an infinite-dimensional vector space over the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, for which no specific basis is known. Consider a basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {b} _{1},\mathbf {b} _{2},\ldots ,\mathbf {b} _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {b} _{1},\mathbf {b} _{2},\ldots ,\mathbf {b} _{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f35afac8a96873cb899fbdc1dfa6663225581b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.804ex; height:2.843ex;" alt="{\displaystyle (\mathbf {b} _{1},\mathbf {b} _{2},\ldots ,\mathbf {b} _{n})}"> of a vector space V of dimension n over a field F. The definition of a basis implies that every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \in V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb70f294c05e671203c3b21e7f5fb3d0b9a0ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.039ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} \in V}"> may be written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =a_{1}\mathbf {b} _{1}+\cdots +a_{n}\mathbf {b} _{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =a_{1}\mathbf {b} _{1}+\cdots +a_{n}\mathbf {b} _{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23433c59a98965eca86b8556cbbbbbb96ca989d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.536ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} =a_{1}\mathbf {b} _{1}+\cdots +a_{n}\mathbf {b} _{n},}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852381be25b656d697c7a4a9634d3dc4c182d833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{n}}"> in F, and that this decomposition is unique. The scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451345cc97e2ed923dd4656fcc400c3f37119cca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\ldots ,a_{n}}"> are called the coordinates of v on the basis. They are also said to be the coefficients of the decomposition of v on the basis. One also says that the n-<a href="/wiki/Tuple" title="Tuple">tuple</a> of the coordinates is the <a href="/wiki/Coordinate_vector" title="Coordinate vector">coordinate vector</a> of v on the basis, since the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </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/caeaf2cf3b94c8a21721478bcd52964068489df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.033ex; height:2.343ex;" alt="{\displaystyle F^{n}}"> of the n-tuples of elements of F is a vector space for <a href="/wiki/Componentwise_operation" class="mw-redirect" title="Componentwise operation">componentwise</a> addition and scalar multiplication, whose dimension is n. The <a href="/wiki/One-to-one_correspondence" class="mw-redirect" title="One-to-one correspondence">one-to-one correspondence</a> between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a <a href="/wiki/Vector_space_isomorphism" class="mw-redirect" title="Vector space isomorphism">vector space isomorphism</a>, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates.<a href="#cite_note-FOOTNOTEHalmos1948[httpsbooksgooglecombooksid1hzYCwAAQBAJpgPA12_12]-20">[17]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=3" title="Edit section: History">edit</a>]</div> Vector spaces stem from <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>, via the introduction of <a href="/wiki/Coordinate" class="mw-redirect" title="Coordinate">coordinates</a> in the plane or three-dimensional space. Around 1636, French mathematicians <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> and <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> founded <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a> by identifying solutions to an equation of two variables with points on a plane <a href="/wiki/Curve" title="Curve">curve</a>.<a href="#cite_note-FOOTNOTEBourbaki1969ch._"Algèbre_linéaire_et_algèbre_multilinéaire",_pp._78–91-21">[18]</a> To achieve geometric solutions without using coordinates, <a href="/wiki/Bernhard_Bolzano" class="mw-redirect" title="Bernhard Bolzano">Bolzano</a> introduced, in 1804, certain operations on points, lines, and planes, which are predecessors of vectors.<a href="#cite_note-FOOTNOTEBolzano1804-22">[19]</a> <a href="#CITEREFMöbius1827">Möbius (1827)</a> introduced the notion of <a href="/wiki/Barycentric_coordinates_(mathematics)" class="mw-redirect" title="Barycentric coordinates (mathematics)">barycentric coordinates</a>.<a href="#cite_note-FOOTNOTEMöbius1827-23">[20]</a> <a href="#CITEREFBellavitis1833">Bellavitis (1833)</a> introduced an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on directed line segments that share the same length and direction which he called <a href="/wiki/Equipollence_(geometry)" title="Equipollence (geometry)">equipollence</a>.<a href="#cite_note-FOOTNOTEBellavitis1833-24">[21]</a> A <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vector</a> is then an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of that relation.<a href="#cite_note-FOOTNOTEDorier1995-25">[22]</a> Vectors were reconsidered with the presentation of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> by <a href="/wiki/Jean-Robert_Argand" title="Jean-Robert Argand">Argand</a> and <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a> and the inception of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> by the latter.<a href="#cite_note-FOOTNOTEHamilton1853-26">[23]</a> They are elements in R2 and R4; treating them using <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> goes back to <a href="/wiki/Laguerre" class="mw-redirect" title="Laguerre">Laguerre</a> in 1867, who also defined <a href="/wiki/System_of_linear_equations" title="System of linear equations">systems of linear equations</a>. In 1857, <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> introduced the <a href="/wiki/Matrix_notation" class="mw-redirect" title="Matrix notation">matrix notation</a> which allows for harmonization and simplification of <a href="/wiki/Linear_map" title="Linear map">linear maps</a>. Around the same time, <a href="/wiki/Grassmann" class="mw-redirect" title="Grassmann">Grassmann</a> studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.<a href="#cite_note-FOOTNOTEGrassmann2000-27">[24]</a> In his work, the concepts of <a href="/wiki/Linear_independence" title="Linear independence">linear independence</a> and <a href="/wiki/Dimension" title="Dimension">dimension</a>, as well as <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar products</a> are present. Grassmann's 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today called <a href="/wiki/Algebras_over_a_field" class="mw-redirect" title="Algebras over a field">algebras</a>. Italian mathematician <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a> was the first to give the modern definition of vector spaces and linear maps in 1888,<a href="#cite_note-FOOTNOTEPeano1888ch._IX-28">[25]</a> although he called them "linear systems".<a href="#cite_note-FOOTNOTEGuo2021-29">[26]</a> Peano's axiomatization allowed for vector spaces with infinite dimension, but Peano did not develop that theory further. In 1897, <a href="/wiki/Salvatore_Pincherle" title="Salvatore Pincherle">Salvatore Pincherle</a> adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.<a href="#cite_note-FOOTNOTEMoore1995268–271-30">[27]</a> An important development of vector spaces is due to the construction of <a href="/wiki/Function_spaces" class="mw-redirect" title="Function spaces">function spaces</a> by <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a>. This was later formalized by <a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach</a> and <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, around 1920.<a href="#cite_note-FOOTNOTEBanach1922-31">[28]</a> At that time, <a href="/wiki/Algebra" title="Algebra">algebra</a> and the new field of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> began to interact, notably with key concepts such as <a href="/wiki/Lp_space" title="Lp space">spaces of p-integrable functions</a> and <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>.<a href="#cite_note-FOOTNOTEDorier1995Moore1995-32">[29]</a> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=4" title="Edit section: Examples">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Examples_of_vector_spaces" title="Examples of vector spaces">Examples of vector spaces</a></div> <div class="mw-heading mw-heading3"><h3 id="Arrows_in_the_plane">Arrows in the plane</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=5" title="Edit section: Arrows in the plane">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:192px;max-width:192px"><div class="trow"><div class="tsingle" style="width:190px;max-width:190px"><div class="thumbimage" style="height:77px;overflow:hidden"><a href="/wiki/File:Vector_addition3.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/188px-Vector_addition3.svg.png" decoding="async" width="188" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/282px-Vector_addition3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Vector_addition3.svg/376px-Vector_addition3.svg.png 2x" data-file-width="190" data-file-height="78" /></a></div><div class="thumbcaption">Vector addition: the sum v + w (black) of the vectors v (blue) and w (red) is shown.</div></div></div><div class="trow"><div class="tsingle" style="width:190px;max-width:190px"><div class="thumbimage" style="height:63px;overflow:hidden"><a href="/wiki/File:Scalar_multiplication.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Scalar_multiplication.svg/188px-Scalar_multiplication.svg.png" decoding="async" width="188" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Scalar_multiplication.svg/282px-Scalar_multiplication.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Scalar_multiplication.svg/376px-Scalar_multiplication.svg.png 2x" data-file-width="312" data-file-height="106" /></a></div><div class="thumbcaption">Scalar multiplication: the multiples −v and 2w are shown.</div></div></div></div></div> The first example of a vector space consists of <a href="/wiki/Arrow_(symbol)" title="Arrow (symbol)">arrows</a> in a fixed <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, starting at one fixed point. This is used in physics to describe <a href="/wiki/Force" title="Force">forces</a> or <a href="/wiki/Velocity" title="Velocity">velocities</a>.<a href="#cite_note-FOOTNOTEKreyszig2020[httpsbooksgooglecombooksidw4T3DwAAQBAJpgPA355_355]-33">[30]</a> Given any two such arrows, v and w, the <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows, and is denoted v + w. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive <a href="/wiki/Real_number" title="Real number">real number</a> a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction instead.<a href="#cite_note-FOOTNOTEKreyszig2020[httpsbooksgooglecombooksidw4T3DwAAQBAJpgPA358_358&ndash;359]-34">[31]</a> The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (the second image). Equivalently, 2w is the sum w + w. Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the second image). <div class="mw-heading mw-heading3"><h3 id="Ordered_pairs_of_numbers">Ordered pairs of numbers</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=6" title="Edit section: Ordered pairs of numbers">edit</a>]</div> A second key example of a vector space is provided by pairs of real numbers x and y. The order of the components x and y is significant, so such a pair is also called an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a>. Such a pair is written as (x, y). The sum of two such pairs and the multiplication of a pair with a number is defined as follows:<a href="#cite_note-FOOTNOTEJain2001[httpsbooksgooglecombooksid-lzAee3uQtICpgPA11_11]-35">[32]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x_{1},y_{1})+(x_{2},y_{2})&=(x_{1}+x_{2},y_{1}+y_{2}),\\a(x,y)&=(ax,ay).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>,</mo> <mi>a</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x_{1},y_{1})+(x_{2},y_{2})&=(x_{1}+x_{2},y_{1}+y_{2}),\\a(x,y)&=(ax,ay).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b70e5ac60990df36550d2cc24b8d3004c608656" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.857ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(x_{1},y_{1})+(x_{2},y_{2})&=(x_{1}+x_{2},y_{1}+y_{2}),\\a(x,y)&=(ax,ay).\end{aligned}}}"> The first example above reduces to this example if an arrow is represented by a pair of <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> of its endpoint. <div class="mw-heading mw-heading3"><h3 id="Coordinate_space">Coordinate space</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=7" title="Edit section: Coordinate space">edit</a>]</div> The simplest example of a vector space over a field F is the field F itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all <a href="/wiki/Tuple" title="Tuple">n-tuples</a> (sequences of length n) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2},\dots ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2},\dots ,a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301ea4e20db36959a961fefe4e3e38a667964d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.038ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2},\dots ,a_{n})}"> of elements ai of F form a vector space that is usually denoted Fn and called a coordinate space.<a href="#cite_note-FOOTNOTELang1987ch._I.1-36">[33]</a> The case n = 1 is the above-mentioned simplest example, in which the field F is also regarded as a vector space over itself. The case F = R and n = 2 (so R2) reduces to the previous example. <div class="mw-heading mw-heading3"><h3 id="Complex_numbers_and_other_field_extensions">Complex numbers and other field extensions</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=8" title="Edit section: Complex numbers and other field extensions">edit</a>]</div> The set of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> C, numbers that can be written in the form x + iy for <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> x and y where i is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, form a vector space over the reals with the usual addition and multiplication: (x + iy) + (a + ib) = (x + a) + i(y + b) and c ⋅ (x + iy) = (c ⋅ x) + i(c ⋅ y) for real numbers x, y, a, b and c. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. The example of complex numbers is essentially the same as (that is, it is isomorphic to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number x + i y as representing the ordered pair (x, y) in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example. More generally, <a href="/wiki/Field_extension" title="Field extension">field extensions</a> provide another class of examples of vector spaces, particularly in algebra and <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>: a field F containing a <a href="/wiki/Field_extension" title="Field extension">smaller field</a> E is an E-vector space, by the given multiplication and addition operations of F.<a href="#cite_note-FOOTNOTELang2002ch._V.1-37">[34]</a> For example, the complex numbers are a vector space over R, and the field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} (i{\sqrt {5}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} (i{\sqrt {5}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79cf13737fd8630f04a1028028c44e95b4729be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.718ex; height:3.009ex;" alt="{\displaystyle \mathbf {Q} (i{\sqrt {5}})}"> is a vector space over Q. <div class="mw-heading mw-heading3"><h3 id="Function_spaces">Function spaces</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=9" title="Edit section: Function spaces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Function_space" title="Function space">Function space</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Example_for_addition_of_functions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Example_for_addition_of_functions.svg/220px-Example_for_addition_of_functions.svg.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Example_for_addition_of_functions.svg/330px-Example_for_addition_of_functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Example_for_addition_of_functions.svg/440px-Example_for_addition_of_functions.svg.png 2x" data-file-width="487" data-file-height="367" /></a><figcaption>Addition of functions: the sum of the sine and the exponential function is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin +\exp :\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>+</mo> <mi>exp</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 \sin +\exp :\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3afcecd5be8d1b1d1f1a990a4a5dac441daa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.898ex; height:2.509ex;" alt="{\displaystyle \sin +\exp :\mathbb {R} \to \mathbb {R} }"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>+</mo> <mi>exp</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc39721fd4825bb41ad0bb72835e65db80e490f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.563ex; height:2.843ex;" alt="{\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)}">.</figcaption></figure> Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18ff99e669a82b98573d50cad35d6d2dc8402b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.044ex; height:2.843ex;" alt="{\displaystyle (f+g)}"> given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(w)=f(w)+g(w),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+g)(w)=f(w)+g(w),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bab026cce85931998fc137b8a02416c02260189" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.445ex; height:2.843ex;" alt="{\displaystyle (f+g)(w)=f(w)+g(w),}"> and similarly for multiplication. Such function spaces occur in many geometric situations, when Ω is the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> or an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>, or other <a href="/wiki/Subset" title="Subset">subsets</a> of R. Many notions in topology and analysis, such as <a href="/wiki/Continuous_function" title="Continuous function">continuity</a>, <a href="/wiki/Integral" title="Integral">integrability</a> or <a href="/wiki/Differentiability" class="mw-redirect" title="Differentiability">differentiability</a> are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.<a href="#cite_note-FOOTNOTELang1993ch._XII.3.,_p._335-38">[35]</a> Therefore, the set of such functions are vector spaces, whose study belongs to <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. <div class="mw-heading mw-heading3"><h3 id="Linear_equations">Linear equations</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=10" title="Edit section: Linear equations">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/Linear_equation" title="Linear equation">Linear equation</a>, <a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear differential equation</a>, and <a href="/wiki/Systems_of_linear_equations" class="mw-redirect" title="Systems of linear equations">Systems of linear equations</a></div> Systems of <a href="/wiki/Homogeneous_linear_equation" class="mw-redirect" title="Homogeneous linear equation">homogeneous linear equations</a> are closely tied to vector spaces.<a href="#cite_note-FOOTNOTELang1987ch._VI.3.-39">[36]</a> For example, the solutions of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{9}&&a\,&&+\,3b\,&\,+&\,&c&\,=0\\4&&a\,&&+\,2b\,&\,+&\,2&c&\,=0\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd /> <mtd /> <mtd> <mi>a</mi> <mspace width="thinmathspace" /> </mtd> <mtd /> <mtd> <mo>+</mo> <mspace width="thinmathspace" /> <mn>3</mn> <mi>b</mi> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <mspace width="thinmathspace" /> <mo>+</mo> </mtd> <mtd> <mspace width="thinmathspace" /> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd /> <mtd> <mi>a</mi> <mspace width="thinmathspace" /> </mtd> <mtd /> <mtd> <mo>+</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mi>b</mi> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <mspace width="thinmathspace" /> <mo>+</mo> </mtd> <mtd> <mspace width="thinmathspace" /> <mn>2</mn> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{9}&&a\,&&+\,3b\,&\,+&\,&c&\,=0\\4&&a\,&&+\,2b\,&\,+&\,2&c&\,=0\\\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aebf9bf8985666ed710395bef8cb0f7ab540634" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.028ex; height:5.843ex;" alt="{\displaystyle {\begin{alignedat}{9}&&a\,&&+\,3b\,&\,+&\,&c&\,=0\\4&&a\,&&+\,2b\,&\,+&\,2&c&\,=0\\\end{alignedat}}}"> are given by triples with arbitrary <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f059f053fcf9f421b7c74362cf3bd5ed024e19d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=a/2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a/2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27f037b8e9aba144fb73ed74634611997b7bd4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.298ex; height:2.843ex;" alt="{\displaystyle b=a/2,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=-5a/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=-5a/2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0b11abd3d89d8d7ff42d1567e1e24459d14f79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.277ex; height:2.843ex;" alt="{\displaystyle c=-5a/2.}"> They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a> can be used to condense multiple linear equations as above into one vector equation, namely <div id="equation3"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} =\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2febae9e52381255ec6072203b6bf65a0c5f611a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.236ex; height:2.509ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {0} ,}"></div> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}1&3&1\\4&2&2\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}1&3&1\\4&2&2\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7540136866058593b6dc72b68d4bb0ca3ba8ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.181ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}1&3&1\\4&2&2\end{bmatrix}}}"> is the matrix containing the coefficients of the given equations, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> is the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b,c),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c052387768871591c5823a1c4ae9c057e95676d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.758ex; height:2.843ex;" alt="{\displaystyle (a,b,c),}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49d802e9b456b1476fcf3e6fd73b0bf266727c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.154ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} }"> denotes the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} =(0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} =(0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015934e10a84cfdabab9ec9b8647f6960dd37dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.603ex; height:2.843ex;" alt="{\displaystyle \mathbf {0} =(0,0)}"> is the zero vector. In a similar vein, the solutions of homogeneous linear differential equations form vector spaces. For example, <div id="equation1"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\prime \prime }(x)+2f^{\prime }(x)+f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\prime \prime }(x)+2f^{\prime }(x)+f(x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6721f2ddd15edfaa7bfc741f6b3cf5cd7ca64407" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.263ex; height:3.009ex;" alt="{\displaystyle f^{\prime \prime }(x)+2f^{\prime }(x)+f(x)=0}"></div> yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ae^{-x}+bxe^{-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> <mo>=</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=ae^{-x}+bxe^{-x},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f878ee6a61dd1ed463a7553069f0cd028c1d52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.629ex; height:3.009ex;" alt="{\displaystyle f(x)=ae^{-x}+bxe^{-x},}"> where <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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> are arbitrary constants, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841c0d168e64191c45a45e54c7e447defd17ec6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}"> is the <a href="/wiki/Natural_exponential_function" class="mw-redirect" title="Natural exponential function">natural exponential function</a>. <div class="mw-heading mw-heading2"><h2 id="Linear_maps_and_matrices">Linear maps and matrices</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=11" title="Edit section: Linear maps and matrices">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_map" title="Linear map">Linear map</a></div> The relation of two vector spaces can be expressed by linear map or <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>. They are <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> that reflect the vector space structure, that is, they preserve sums and scalar multiplication: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(\mathbf {v} +\mathbf {w} )&=f(\mathbf {v} )+f(\mathbf {w} ),\\f(a\cdot \mathbf {v} )&=a\cdot f(\mathbf {v} )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(\mathbf {v} +\mathbf {w} )&=f(\mathbf {v} )+f(\mathbf {w} ),\\f(a\cdot \mathbf {v} )&=a\cdot f(\mathbf {v} )\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9681922d7136c6d4c62fa80cbe835e6565c091e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.126ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}f(\mathbf {v} +\mathbf {w} )&=f(\mathbf {v} )+f(\mathbf {w} ),\\f(a\cdot \mathbf {v} )&=a\cdot f(\mathbf {v} )\end{aligned}}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"> in <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"> all <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}"> in <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b34655c2ef19b56c81af0e6d1f2f6df0d3ed33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle F.}"><a href="#cite_note-FOOTNOTERoman2005ch._2,_p._45-40">[37]</a> An <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> is a linear map f : V → W such that there exists an <a href="/wiki/Inverse_map" class="mw-redirect" title="Inverse map">inverse map</a> g : W → V, which is a map such that the two possible <a href="/wiki/Function_composition" title="Function composition">compositions</a> f ∘ g : W → W and g ∘ f : V → V are <a href="/wiki/Identity_function" title="Identity function">identity maps</a>. Equivalently, f is both one-to-one (<a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>) and onto (<a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>).<a href="#cite_note-FOOTNOTELang1987ch._IV.4,_Corollary,_p._106-41">[38]</a> If there exists an isomorphism between V and W, the two spaces are said to be isomorphic; they are then essentially identical as vector spaces, since all identities holding in V are, via f, transported to similar ones in W, and vice versa via g. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vector_components.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Vector_components.svg/180px-Vector_components.svg.png" decoding="async" width="180" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Vector_components.svg/270px-Vector_components.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Vector_components.svg/360px-Vector_components.svg.png 2x" data-file-width="196" data-file-height="154" /></a><figcaption>Describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces.</figcaption></figure> For example, the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above (see <a href="#Examples">§ Examples</a>) are isomorphic: a planar arrow v departing at the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> of some (fixed) <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> can be expressed as an ordered pair by considering the x- and y-component of the arrow, as shown in the image at the right. Conversely, given a pair (x, y), the arrow going by x to the right (or to the left, if x is negative), and y up (down, if y is negative) turns back the arrow v.<a href="#cite_note-FOOTNOTENicholson2018ch._7.3-42">[39]</a> Linear maps V → W between two vector spaces form a vector space HomF(V, W), also denoted L(V, W), or 𝓛(V, W).<a href="#cite_note-FOOTNOTELang1987Example_IV.2.6-43">[40]</a> The space of linear maps from V to F is called the <a href="/wiki/Dual_vector_space" class="mw-redirect" title="Dual vector space">dual vector space</a>, denoted V∗.<a href="#cite_note-FOOTNOTELang1987ch._VI.6-44">[41]</a> Via the injective <a href="/wiki/Natural_(category_theory)" class="mw-redirect" title="Natural (category theory)">natural</a> map V → V∗∗, any vector space can be embedded into its bidual; the map is an isomorphism if and only if the space is finite-dimensional.<a href="#cite_note-FOOTNOTEHalmos1974p._28,_Ex._9-45">[42]</a> Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.<a href="#cite_note-FOOTNOTELang1987Theorem_IV.2.1,_p._95-46">[43]</a> If dim V = dim W, a <a href="/wiki/Bijection" title="Bijection">1-to-1 correspondence</a> between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition.<a href="#cite_note-FOOTNOTERoman2005Th._2.5_and_2.6,_p._49-47">[44]</a> Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field is completely classified (<a href="/wiki/Up_to" title="Up to">up to</a> isomorphism) by its dimension, a single number. In particular, any n-dimensional F-vector space V is isomorphic to Fn. However, there is no "canonical" or preferred isomorphism; an isomorphism φ : Fn → V is equivalent to the choice of a basis of V, by mapping the standard basis of Fn to V, via φ. <div class="mw-heading mw-heading3"><h3 id="Matrices">Matrices</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=12" title="Edit section: Matrices">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/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a> and <a href="/wiki/Determinant" title="Determinant">Determinant</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Matrix.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Matrix.svg/200px-Matrix.svg.png" decoding="async" width="200" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Matrix.svg/300px-Matrix.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Matrix.svg/400px-Matrix.svg.png 2x" data-file-width="224" data-file-height="189" /></a><figcaption>A typical matrix</figcaption></figure> Matrices are a useful notion to encode linear maps.<a href="#cite_note-FOOTNOTELang1987ch._V.1-48">[45]</a> They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix <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}"> gives rise to a linear map from Fn to Fm, by the following <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\mapsto \left(\sum _{j=1}^{n}a_{1j}x_{j},\sum _{j=1}^{n}a_{2j}x_{j},\ldots ,\sum _{j=1}^{n}a_{mj}x_{j}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\mapsto \left(\sum _{j=1}^{n}a_{1j}x_{j},\sum _{j=1}^{n}a_{2j}x_{j},\ldots ,\sum _{j=1}^{n}a_{mj}x_{j}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c4e62a6ac16d4156d7fccadb5bf0346e1bf4a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:61.838ex; height:7.676ex;" alt="{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\mapsto \left(\sum _{j=1}^{n}a_{1j}x_{j},\sum _{j=1}^{n}a_{2j}x_{j},\ldots ,\sum _{j=1}^{n}a_{mj}x_{j}\right),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2b0b7618be940f4e8c0d27f05ab75fbc13e83c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.454ex; height:2.843ex;" alt="{\textstyle \sum }"> denotes <a href="/wiki/Summation" title="Summation">summation</a>, or by using the <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> of the matrix <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}"> with the coordinate vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }">: <div id="equation2"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \mapsto A\mathbf {x} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">↦</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \mapsto A\mathbf {x} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8e51a6109f62db762d60aa65e7fd6b0266b0fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.826ex; height:2.176ex;" alt="{\displaystyle \mathbf {x} \mapsto A\mathbf {x} .}"></div> Moreover, after choosing bases of V and W, any linear map f : V → W is uniquely represented by a matrix via this assignment.<a href="#cite_note-FOOTNOTELang1987ch._V.3.,_Corollary,_p._106-49">[46]</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Determinant_parallelepiped.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Determinant_parallelepiped.svg/200px-Determinant_parallelepiped.svg.png" decoding="async" width="200" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Determinant_parallelepiped.svg/300px-Determinant_parallelepiped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Determinant_parallelepiped.svg/400px-Determinant_parallelepiped.svg.png 2x" data-file-width="950" data-file-height="800" /></a><figcaption>The volume of this <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a> is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r1, r2, and r3.</figcaption></figure> The <a href="/wiki/Determinant" title="Determinant">determinant</a> det (A) of a <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.<a href="#cite_note-FOOTNOTELang1987Theorem_VII.9.8,_p._198-50">[47]</a> The linear transformation of Rn corresponding to a real n-by-n matrix is <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">orientation preserving</a> if and only if its determinant is positive. <div class="mw-heading mw-heading3"><h3 id="Eigenvalues_and_eigenvectors">Eigenvalues and eigenvectors</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=13" title="Edit section: Eigenvalues and eigenvectors">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></div> <a href="/wiki/Endomorphism" title="Endomorphism">Endomorphisms</a>, linear maps f : V → V, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying λv = f(v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ.<a href="#cite_note-FOOTNOTERoman2005ch._8,_p._135–156-51">[48]</a> Equivalently, v is an element of the <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a> of the difference f − λ · Id (where Id is the <a href="/wiki/Identity_function" title="Identity function">identity map</a> V → V). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(f-\lambda \cdot \operatorname {Id} )=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>λ</mi> <mo>⋅</mo> <mi>Id</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(f-\lambda \cdot \operatorname {Id} )=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d369d3a0465cb4b54b0f9c37f0e403e39daa31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.232ex; height:2.843ex;" alt="{\displaystyle \det(f-\lambda \cdot \operatorname {Id} )=0.}"> By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in λ, called the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of f.<a href="#cite_note-FOOTNOTELang1987ch._IX.4-52">[49]</a> If the field F is large enough to contain a zero of this polynomial (which automatically happens for F <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a>, such as F = C) any linear map has at least one eigenvector. The vector space V may or may not possess an <a href="/wiki/Eigenbasis" class="mw-redirect" title="Eigenbasis">eigenbasis</a>, a basis consisting of eigenvectors. This phenomenon is governed by the <a href="/wiki/Jordan_canonical_form" class="mw-redirect" title="Jordan canonical form">Jordan canonical form</a> of the map.<a href="#cite_note-FOOTNOTERoman2005ch._8,_p._140-53">[50]</a> The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in question. <div class="mw-heading mw-heading2"><h2 id="Basic_constructions">Basic constructions</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=14" title="Edit section: Basic constructions">edit</a>]</div> In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. <div class="mw-heading mw-heading3"><h3 id="Subspaces_and_quotient_spaces">Subspaces and quotient spaces</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=15" title="Edit section: Subspaces and quotient spaces">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/Linear_subspace" title="Linear subspace">Linear subspace</a> and <a href="/wiki/Quotient_vector_space" class="mw-redirect" title="Quotient vector space">Quotient vector space</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Linear_subspaces_with_shading.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/250px-Linear_subspaces_with_shading.svg.png" decoding="async" width="250" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/375px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/500px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a><figcaption>A line passing through the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> (blue, thick) in <a href="/wiki/Euclidean_space" title="Euclidean space">R3</a> is a linear subspace. It is the intersection of two <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">planes</a> (green and yellow).</figcaption></figure> A nonempty <a href="/wiki/Subset" title="Subset">subset</a> <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 a vector space <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}"> that is closed under addition and scalar multiplication (and therefore contains the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }">-vector of <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}">) is called a linear subspace of <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}">, or simply a subspace of <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}">, when the ambient space is unambiguously a vector space.<a href="#cite_note-FOOTNOTERoman2005ch._1,_p._29-54">[51]</a><a href="#cite_note-55">[nb 4]</a> Subspaces of <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}"> are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given 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}"> of vectors is called its <a href="/wiki/Linear_span" title="Linear span">span</a>, and it is the smallest subspace of <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}"> containing the 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}">. Expressed in terms of elements, the span is the subspace consisting of all the <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of elements 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}">.<a href="#cite_note-FOOTNOTERoman2005ch._1,_p._35-56">[52]</a> Linear subspace of dimension 1 and 2 are referred to as a line (also vector line), and a plane respectively. If W is an n-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-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-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> is called a <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a>.<a href="#cite_note-FOOTNOTENicholson2018ch._10.4-57">[53]</a> The counterpart to subspaces are quotient vector spaces.<a href="#cite_note-FOOTNOTERoman2005ch._3,_p._64-58">[54]</a> Given any subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\subseteq V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>⊆</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\subseteq V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d05af9d62859741b09533bd82087feb0a9b298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.321ex; height:2.343ex;" alt="{\displaystyle W\subseteq V}">, the quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V/W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V/W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74f0444abe38d7c9392ba2cfd13031b0f3d39cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.843ex;" alt="{\displaystyle V/W}"> ("<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}"> <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> <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}">") is defined as follows: as a set, it consists of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} +W=\{\mathbf {v} +\mathbf {w} :\mathbf {w} \in W\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mi>W</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>∈</mo> <mi>W</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} +W=\{\mathbf {v} +\mathbf {w} :\mathbf {w} \in W\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f7a517f4b3041c24da6996b220fc1c86cc4f438" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.084ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} +W=\{\mathbf {v} +\mathbf {w} :\mathbf {w} \in W\},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> is an arbitrary vector in <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}">. The sum of two such elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}+W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}+W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afad24e3047dd467c6b0d71d1cb265a21dd14de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.741ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} _{1}+W}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{2}+W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{2}+W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93a1952bd05dcb8f2c1bacc64785015b2d020f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.741ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} _{2}+W}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {v} _{1}+\mathbf {v} _{2}\right)+W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {v} _{1}+\mathbf {v} _{2}\right)+W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01b2554e57d9cab67b748eb44ed44b90fdd1216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.856ex; height:2.843ex;" alt="{\displaystyle \left(\mathbf {v} _{1}+\mathbf {v} _{2}\right)+W}">, and scalar multiplication is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (\mathbf {v} +W)=(a\cdot \mathbf {v} )+W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (\mathbf {v} +W)=(a\cdot \mathbf {v} )+W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e97c4a5f8414686cc85e025dde11d1c10f70e4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.908ex; height:2.843ex;" alt="{\displaystyle a\cdot (\mathbf {v} +W)=(a\cdot \mathbf {v} )+W}">. The key point in this definition is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}+W=\mathbf {v} _{2}+W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>W</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}+W=\mathbf {v} _{2}+W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ba99b77b0d83e0af1328f84336e84d0d18b21f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.58ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} _{1}+W=\mathbf {v} _{2}+W}"> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the difference of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282458bb19c231f94697405bddd93af04a34cabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498720fbe6f897f2b86d2cf0f37498d682932aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{2}}"> lies in <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}">.<a href="#cite_note-59">[nb 5]</a> This way, the quotient space "forgets" information that is contained in the subspace <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}">. The <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d2eca3185b45d4462e15bf85bd4ad36e32062d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \ker(f)}"> of a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\to W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574dffa1c85efaef6b6ef553ebd8ad9cf7f87fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\to W}"> consists of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> that are mapped to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"> in <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}">.<a href="#cite_note-FOOTNOTELang1987ch._IV.3.-60">[55]</a> The kernel and the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {im} (f)=\{f(\mathbf {v} ):\mathbf {v} \in V\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {im} (f)=\{f(\mathbf {v} ):\mathbf {v} \in V\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c17f1082d2ffdee338c8250f2f88011407b458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.569ex; height:2.843ex;" alt="{\displaystyle \operatorname {im} (f)=\{f(\mathbf {v} ):\mathbf {v} \in V\}}"> are subspaces of <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}"> and <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}">, respectively.<a href="#cite_note-FOOTNOTERoman2005ch._2,_p._48-61">[56]</a> An important example is the kernel of a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \mapsto A\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">↦</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \mapsto A\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5a1063a4f66331d643d57acd21b22bc77d009d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.179ex; height:2.176ex;" alt="{\displaystyle \mathbf {x} \mapsto A\mathbf {x} }"> for some fixed matrix <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}">. The kernel of this map is the subspace of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f51dab2d10f98cebc72b24960edf3e14e06c1e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.589ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {0} }">, which is precisely the set of solutions to the system of homogeneous linear equations belonging to <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}">. This concept also extends to linear differential equations <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}f+a_{1}{\frac {df}{dx}}+a_{2}{\frac {d^{2}f}{dx^{2}}}+\cdots +a_{n}{\frac {d^{n}f}{dx^{n}}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>f</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}f+a_{1}{\frac {df}{dx}}+a_{2}{\frac {d^{2}f}{dx^{2}}}+\cdots +a_{n}{\frac {d^{n}f}{dx^{n}}}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8237f82c63139d443a99b7a3873373a25953760b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.989ex; height:6.009ex;" alt="{\displaystyle a_{0}f+a_{1}{\frac {df}{dx}}+a_{2}{\frac {d^{2}f}{dx^{2}}}+\cdots +a_{n}{\frac {d^{n}f}{dx^{n}}}=0,}"> where the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> are functions 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/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,}"> too. In the corresponding map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto D(f)=\sum _{i=0}^{n}a_{i}{\frac {d^{i}f}{dx^{i}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto D(f)=\sum _{i=0}^{n}a_{i}{\frac {d^{i}f}{dx^{i}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64f7f29f1c6a4d5e8fae8d8a7ff8520bb864c29b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.603ex; height:6.843ex;" alt="{\displaystyle f\mapsto D(f)=\sum _{i=0}^{n}a_{i}{\frac {d^{i}f}{dx^{i}}},}"> the <a href="/wiki/Derivative" title="Derivative">derivatives</a> of the function <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}"> appear linearly (as opposed to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\prime \prime }(x)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\prime \prime }(x)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad191043d35e0e30f217b0cc79b27754309cc0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.651ex; height:3.176ex;" alt="{\displaystyle f^{\prime \prime }(x)^{2}}">, for example). Since differentiation is a linear procedure (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)^{\prime }=f^{\prime }+g^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+g)^{\prime }=f^{\prime }+g^{\prime }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bd3fffda890ce19fcbf3ebbf5adffc29aaeb51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.476ex; height:3.009ex;" alt="{\displaystyle (f+g)^{\prime }=f^{\prime }+g^{\prime }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c\cdot f)^{\prime }=c\cdot f^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>⋅</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c\cdot f)^{\prime }=c\cdot f^{\prime }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8aa02e2100aa5a1ab0d2273325f9cdd04763106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.248ex; height:3.009ex;" alt="{\displaystyle (c\cdot f)^{\prime }=c\cdot f^{\prime }}"> for a constant <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}">) this assignment is linear, called a <a href="/wiki/Linear_differential_operator" class="mw-redirect" title="Linear differential operator">linear differential operator</a>. In particular, the solutions to the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(f)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(f)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5e2fc2c1dc63439f0f204f447fe79bf4d6bdde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.273ex; height:2.843ex;" alt="{\displaystyle D(f)=0}"> form a vector space (over R or C).<a href="#cite_note-FOOTNOTENicholson2018ch._7.4-62">[57]</a> The existence of kernels and images is part of the statement that the <a href="/wiki/Category_of_vector_spaces" class="mw-redirect" title="Category of vector spaces">category of vector spaces</a> (over a fixed field <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}">) is an <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a>, that is, a corpus of mathematical objects and structure-preserving maps between them (a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>) that behaves much like the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a>.<a href="#cite_note-FOOTNOTEMac_Lane1998-63">[58]</a> Because of this, many statements such as the <a href="/wiki/First_isomorphism_theorem" class="mw-redirect" title="First isomorphism theorem">first isomorphism theorem</a> (also called <a href="/wiki/Rank%E2%80%93nullity_theorem" title="Rank–nullity theorem">rank–nullity theorem</a> in matrix-related terms) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V/\ker(f)\;\equiv \;\operatorname {im} (f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>≡</mo> <mspace width="thickmathspace" /> <mi>im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V/\ker(f)\;\equiv \;\operatorname {im} (f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1ede3953889f91ee4386a3a6ccbc25f2a209e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.656ex; height:2.843ex;" alt="{\displaystyle V/\ker(f)\;\equiv \;\operatorname {im} (f)}"> and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>. <div class="mw-heading mw-heading3"><h3 id="Direct_product_and_direct_sum">Direct product and direct sum</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=16" title="Edit section: Direct product and direct sum">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/Direct_product" title="Direct product">Direct product</a> and <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">Direct sum of modules</a></div> The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. The direct product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\prod _{i\in I}V_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\prod _{i\in I}V_{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7276b7a71c0022b5221c0a2a891548b1b01159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.46ex; height:3.009ex;" alt="{\displaystyle \textstyle {\prod _{i\in I}V_{i}}}"> of a family of vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}"> consists of the set of all tuples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {v} _{i}\right)_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <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 \left(\mathbf {v} _{i}\right)_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51b0c3b6453355a143a01f1222f310c7bb4dd9d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.744ex; height:3.009ex;" alt="{\displaystyle \left(\mathbf {v} _{i}\right)_{i\in I}}">, which specify for each 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> </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}"> in some <a href="/wiki/Index_set" title="Index set">index set</a> <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/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}">.<a href="#cite_note-FOOTNOTERoman2005ch._1,_pp._31–32-64">[59]</a> Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigoplus _{i\in I}V_{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> <mi>I</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigoplus _{i\in I}V_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792fd905a6c6c21b7e00e81a0b785840edc59b2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.848ex; height:3.009ex;" alt="{\textstyle \bigoplus _{i\in I}V_{i}}"> (also called <a href="/wiki/Coproduct" title="Coproduct">coproduct</a> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \coprod _{i\in I}V_{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> <mi>I</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \coprod _{i\in I}V_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf46cb4bf90baee84164db1246e0018346c2df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.46ex; height:3.009ex;" alt="{\textstyle \coprod _{i\in I}V_{i}}">), where only tuples with finitely many nonzero vectors are allowed. If the index set <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/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is finite, the two constructions agree, but in general they are different. <div class="mw-heading mw-heading3"><h3 id="Tensor_product">Tensor product</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=17" title="Edit section: Tensor product">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Tensor_product_of_vector_spaces" class="mw-redirect" title="Tensor product of vector spaces">Tensor product of vector spaces</a></div> The tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\otimes _{F}W,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mi>W</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes _{F}W,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94bed517f45b4e6ecfe0df9a3862d5c895be220a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.173ex; height:2.509ex;" alt="{\displaystyle V\otimes _{F}W,}"> or simply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\otimes W,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊗</mo> <mi>W</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes W,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c573e8e21370a278ec3c8f0225c1f970fe583eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.71ex; height:2.509ex;" alt="{\displaystyle V\otimes W,}"> of two vector spaces <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}"> and <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}"> is one of the central notions of <a href="/wiki/Multilinear_algebra" title="Multilinear algebra">multilinear algebra</a> which deals with extending notions such as linear maps to several variables. A map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:V\times W\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>V</mi> <mo>×</mo> <mi>W</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:V\times W\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573030f61b9c05c1a6d6f421c3a6aaf9dffe8f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.71ex; height:2.509ex;" alt="{\displaystyle g:V\times W\to X}"> from the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\times W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>×</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\times W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9b65a794c51f2c6b0396c9d01d3bbe48f54fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.063ex; height:2.176ex;" alt="{\displaystyle V\times W}"> is called <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> is linear in both variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf8a103f40a7fbdb19c7e667d92541b26cae840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.578ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} .}"> That is to say, for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"> the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \mapsto g(\mathbf {v} ,\mathbf {w} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">↦</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \mapsto g(\mathbf {v} ,\mathbf {w} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0161f16d0b376773ad41a63192e7f768e921b0f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.326ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} \mapsto g(\mathbf {v} ,\mathbf {w} )}"> is linear in the sense above and likewise for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403a39746fb7416e8917e81d974df3d75b710131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.058ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} .}"> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Universal_tensor_prod.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Universal_tensor_prod.svg/200px-Universal_tensor_prod.svg.png" decoding="async" width="200" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Universal_tensor_prod.svg/300px-Universal_tensor_prod.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Universal_tensor_prod.svg/400px-Universal_tensor_prod.svg.png 2x" data-file-width="248" data-file-height="142" /></a><figcaption><a href="/wiki/Commutative_diagram" title="Commutative diagram">Commutative diagram</a> depicting the universal property of the tensor product</figcaption></figure> The tensor product is a particular vector space that is a universal recipient of bilinear maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f2986cd965e404a1ee33ec84baee5c43da47fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g,}"> as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called <a href="/wiki/Tensor" title="Tensor">tensors</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {w} _{1}+\mathbf {v} _{2}\otimes \mathbf {w} _{2}+\cdots +\mathbf {v} _{n}\otimes \mathbf {w} _{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}\otimes \mathbf {w} _{1}+\mathbf {v} _{2}\otimes \mathbf {w} _{2}+\cdots +\mathbf {v} _{n}\otimes \mathbf {w} _{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e928ff3fc8739d57c876a0596c0c69d51f5043f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.093ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {w} _{1}+\mathbf {v} _{2}\otimes \mathbf {w} _{2}+\cdots +\mathbf {v} _{n}\otimes \mathbf {w} _{n},}"> subject to the rules<a href="#cite_note-FOOTNOTELang2002ch._XVI.1-65">[60]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{6}a\cdot (\mathbf {v} \otimes \mathbf {w} )~&=~(a\cdot \mathbf {v} )\otimes \mathbf {w} ~=~\mathbf {v} \otimes (a\cdot \mathbf {w} ),&&~~{\text{ where }}a{\text{ is a scalar}}\\(\mathbf {v} _{1}+\mathbf {v} _{2})\otimes \mathbf {w} ~&=~\mathbf {v} _{1}\otimes \mathbf {w} +\mathbf {v} _{2}\otimes \mathbf {w} &&\\\mathbf {v} \otimes (\mathbf {w} _{1}+\mathbf {w} _{2})~&=~\mathbf {v} \otimes \mathbf {w} _{1}+\mathbf {v} \otimes \mathbf {w} _{2}.&&\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a scalar</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mtd> <mtd /> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{6}a\cdot (\mathbf {v} \otimes \mathbf {w} )~&=~(a\cdot \mathbf {v} )\otimes \mathbf {w} ~=~\mathbf {v} \otimes (a\cdot \mathbf {w} ),&&~~{\text{ where }}a{\text{ is a scalar}}\\(\mathbf {v} _{1}+\mathbf {v} _{2})\otimes \mathbf {w} ~&=~\mathbf {v} _{1}\otimes \mathbf {w} +\mathbf {v} _{2}\otimes \mathbf {w} &&\\\mathbf {v} \otimes (\mathbf {w} _{1}+\mathbf {w} _{2})~&=~\mathbf {v} \otimes \mathbf {w} _{1}+\mathbf {v} \otimes \mathbf {w} _{2}.&&\\\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e663ae7296449dad89523a3d8fe5f2902fcf31c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:66.39ex; height:9.176ex;" alt="{\displaystyle {\begin{alignedat}{6}a\cdot (\mathbf {v} \otimes \mathbf {w} )~&=~(a\cdot \mathbf {v} )\otimes \mathbf {w} ~=~\mathbf {v} \otimes (a\cdot \mathbf {w} ),&&~~{\text{ where }}a{\text{ is a scalar}}\\(\mathbf {v} _{1}+\mathbf {v} _{2})\otimes \mathbf {w} ~&=~\mathbf {v} _{1}\otimes \mathbf {w} +\mathbf {v} _{2}\otimes \mathbf {w} &&\\\mathbf {v} \otimes (\mathbf {w} _{1}+\mathbf {w} _{2})~&=~\mathbf {v} \otimes \mathbf {w} _{1}+\mathbf {v} \otimes \mathbf {w} _{2}.&&\\\end{alignedat}}}"> These rules ensure that the map <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}"> from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\times W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>×</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\times W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9b65a794c51f2c6b0396c9d01d3bbe48f54fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.063ex; height:2.176ex;" alt="{\displaystyle V\times W}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\otimes W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊗</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d8e48e05a95d9c68f80f49e3d509ba9de064c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.063ex; height:2.343ex;" alt="{\displaystyle V\otimes W}"> that maps a <a href="/wiki/Tuple" title="Tuple">tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {v} ,\mathbf {w} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {v} ,\mathbf {w} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbbf33643a4bc2da3a392eb0af42e1b519afcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.185ex; height:2.843ex;" alt="{\displaystyle (\mathbf {v} ,\mathbf {w} )}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \otimes \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \otimes \mathbf {w} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b446f6a0aada1823198d4442ec0a7244b9ae255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.183ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} \otimes \mathbf {w} }"> is bilinear. The universality states that given 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 any bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:V\times W\to X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>V</mi> <mo>×</mo> <mi>W</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:V\times W\to X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1024452187992063c0bc54fceeed52a1ce9f37ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.357ex; height:2.509ex;" alt="{\displaystyle g:V\times W\to X,}"> there exists a unique map <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,}"> shown in the diagram with a dotted arrow, whose <a href="/wiki/Function_composition" title="Function composition">composition</a> with <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}"> equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68215c5fb5297aa27b47386a65b30831e4584b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.408ex; height:2.009ex;" alt="{\displaystyle g:}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\mathbf {v} \otimes \mathbf {w} )=g(\mathbf {v} ,\mathbf {w} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(\mathbf {v} \otimes \mathbf {w} )=g(\mathbf {v} ,\mathbf {w} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7c0da737ebb45048abdcfa507abb0cd22c729a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.368ex; height:2.843ex;" alt="{\displaystyle u(\mathbf {v} \otimes \mathbf {w} )=g(\mathbf {v} ,\mathbf {w} ).}"><a href="#cite_note-66">[61]</a> This is called the <a href="/wiki/Universal_property" title="Universal property">universal property</a> of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object. <div class="mw-heading mw-heading2"><h2 id="Vector_spaces_with_additional_structure">Vector spaces with additional structure</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=18" title="Edit section: Vector spaces with additional structure">edit</a>]</div> From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converges</a> to another function. Likewise, linear algebra is not adapted to deal with <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>, since the addition operation allows only finitely many terms to be added. Therefore, the needs of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> require considering additional structures.<a href="#cite_note-FOOTNOTERudin1991p.3-67">[62]</a> A vector space may be given a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\leq ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>≤</mo> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\leq ,\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b5304f5dfd5b1b91c2c52b32a03fc82b7f4a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.229ex; height:2.343ex;" alt="{\displaystyle \,\leq ,\,}"> under which some vectors can be compared.<a href="#cite_note-FOOTNOTESchaeferWolff1999pp._204–205-68">[63]</a> For example, <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 real space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed657d7f0d7aa156e7b9b171f22b4a3aa6482c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n}}"> can be ordered by comparing its vectors componentwise. <a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector spaces</a>, for example <a href="/wiki/Riesz_space" title="Riesz space">Riesz spaces</a>, are fundamental to <a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a>, which relies on the ability to express a function as a difference of two positive functions <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=f^{+}-f^{-}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=f^{+}-f^{-}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784d56e3b9cbfaaae5e1cc982983f8eb91f56761" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.527ex; height:2.843ex;" alt="{\displaystyle f=f^{+}-f^{-}.}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1287ce7b0d53ec4a6c26a4c31f5a8f747be9ba98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.831ex; height:2.843ex;" alt="{\displaystyle f^{+}}"> denotes the positive part of <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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c160282cc6444329bc1a54ab6e1f6c6530c594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.831ex; height:2.843ex;" alt="{\displaystyle f^{-}}"> the negative part.<a href="#cite_note-FOOTNOTEBourbaki2004ch._2,_p._48-69">[64]</a> <div class="mw-heading mw-heading3"><h3 id="Normed_vector_spaces_and_inner_product_spaces">Normed vector spaces and inner product spaces</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=19" title="Edit section: Normed vector spaces and inner product spaces">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/Normed_vector_space" title="Normed vector space">Normed vector space</a> and <a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></div> "Measuring" vectors is done by specifying a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, a datum which measures lengths of vectors, or by an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a>, which measures angles between vectors. Norms and inner products are denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {v} |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {v} |}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a779e65de92152d395f5576ce1001c8e56d7f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.705ex; height:2.843ex;" alt="{\displaystyle |\mathbf {v} |}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4346bdf1bd9b9613acc20abb4389b6ac1d260f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.832ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle ,}"> respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |\mathbf {v} |:={\sqrt {\langle \mathbf {v} ,\mathbf {v} \rangle }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |\mathbf {v} |:={\sqrt {\langle \mathbf {v} ,\mathbf {v} \rangle }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f614a517b5cbf88c3375b1430f049c87aec79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.086ex; height:3.343ex;" alt="{\textstyle |\mathbf {v} |:={\sqrt {\langle \mathbf {v} ,\mathbf {v} \rangle }}.}"> Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.<a href="#cite_note-FOOTNOTERoman2005ch._9-70">[65]</a> Coordinate space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </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/caeaf2cf3b94c8a21721478bcd52964068489df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.033ex; height:2.343ex;" alt="{\displaystyle F^{n}}"> can be equipped with the standard <a href="/wiki/Dot_product" title="Dot product">dot product</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} \cdot \mathbf {y} =x_{1}y_{1}+\cdots +x_{n}y_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} \cdot \mathbf {y} =x_{1}y_{1}+\cdots +x_{n}y_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ccff40a4dd85b183730657b9874e0de84ad2cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.897ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} \cdot \mathbf {y} =x_{1}y_{1}+\cdots +x_{n}y_{n}.}"> In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec417570cf838238bb2d86c60d2cbfab974607cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.704ex; height:3.009ex;" alt="{\displaystyle \mathbf {R} ^{2},}"> this reflects the common notion of the angle between two vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcd6039f384fd6a570c06144b7fc365111eac83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} ,}"> by the <a href="/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\cos \left(\angle (\mathbf {x} ,\mathbf {y} )\right)\cdot |\mathbf {x} |\cdot |\mathbf {y} |.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∠</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \cdot \mathbf {y} =\cos \left(\angle (\mathbf {x} ,\mathbf {y} )\right)\cdot |\mathbf {x} |\cdot |\mathbf {y} |.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3aebaa2199918778e31afd4e84b132d223dc6ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.278ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\cos \left(\angle (\mathbf {x} ,\mathbf {y} )\right)\cdot |\mathbf {x} |\cdot |\mathbf {y} |.}"> Because of this, two vectors satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c94233efe3dbc09548f4716e5c44438c7ea8a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.926ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}"> are called <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a>. An important variant of the standard dot product is used in <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd186b8abaeed43c225609f9556d8ee783b6a7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{4}}"> endowed with the Lorentz product<a href="#cite_note-FOOTNOTENaber2003ch._1.2-71">[66]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0613e2ebaa3b76e545aaeeb8b8b7dfe2ddac4a89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.854ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.}"> In contrast to the standard dot product, it is not <a href="/wiki/Positive_definite_bilinear_form" class="mw-redirect" title="Positive definite bilinear form">positive definite</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {x} |\mathbf {x} \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {x} |\mathbf {x} \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c473743b12e418662333624c8a95e4f475e89e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.278ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {x} |\mathbf {x} \rangle }"> also takes negative values, for example, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =(0,0,0,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(0,0,0,1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/152aae0b6d32f1144e8839d98002c7d2778b145f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.717ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =(0,0,0,1).}"> Singling out the fourth coordinate—<a href="/wiki/Timelike" class="mw-redirect" title="Timelike">corresponding to time</a>, as opposed to three space-dimensions—makes it useful for the mathematical treatment of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. Note that in other conventions time is often written as the first, or "zeroeth" component so that the Lorentz product is written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed2585c032964a80f3534e345d38dc06d98b4a97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.662ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}"> <div class="mw-heading mw-heading3"><h3 id="Topological_vector_spaces">Topological vector spaces</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=20" title="Edit section: Topological vector spaces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector space</a></div> Convergence questions are treated by considering vector spaces <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}"> carrying a compatible <a href="/wiki/Topological_space" title="Topological space">topology</a>, a structure that allows one to talk about elements being <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">close to each other</a>.<a href="#cite_note-FOOTNOTETreves1967Bourbaki1987-72">[67]</a> Compatible here means that addition and scalar multiplication have to be <a href="/wiki/Continuous_map" class="mw-redirect" title="Continuous map">continuous maps</a>. Roughly, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25a040b592282dc2a254c3117e792c3c81161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }"> in <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}">, 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}"> in <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> vary by a bounded amount, then so do <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} +\mathbf {y} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} +\mathbf {y} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489fe3f87c1bac975b1bdea1a3e7b0369d8f7a08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.662ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} +\mathbf {y} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mathbf {x} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mathbf {x} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecda72965b1e05af5802cb3ac8cf540ff513397e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.288ex; height:1.676ex;" alt="{\displaystyle a\mathbf {x} .}"><a href="#cite_note-73">[nb 6]</a> To make sense of specifying the amount a scalar changes, the field <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> also has to carry a topology in this context; a common choice is the reals or the complex numbers. In such topological vector spaces one can consider <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> of vectors. The <a href="/wiki/Infinite_sum" class="mw-redirect" title="Infinite sum">infinite sum</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{\infty }f_{i}~=~\lim _{n\to \infty }f_{1}+\cdots +f_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" 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>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{\infty }f_{i}~=~\lim _{n\to \infty }f_{1}+\cdots +f_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49332ebd615bf4140938529f15b3ed01e1b10513" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.943ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{\infty }f_{i}~=~\lim _{n\to \infty }f_{1}+\cdots +f_{n}}"> denotes the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a> of the corresponding finite partial sums of the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},f_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1},f_{2},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8014204900bd325a7519524aff99e3dc8114fd8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.178ex; height:2.509ex;" alt="{\displaystyle f_{1},f_{2},\ldots }"> of elements of <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}"> For example, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"> could be (real or complex) functions belonging to some <a href="/wiki/Function_space" title="Function space">function space</a> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"> in which case the series is a <a href="/wiki/Function_series" title="Function series">function series</a>. The <a href="/wiki/Modes_of_convergence" title="Modes of convergence">mode of convergence</a> of the series depends on the topology imposed on the function space. In such cases, <a href="/wiki/Pointwise_convergence" title="Pointwise convergence">pointwise convergence</a> and <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a> are two prominent examples.<a href="#cite_note-FOOTNOTESchaeferWolff1999p._7-74">[68]</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vector_norms2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Vector_norms2.svg/250px-Vector_norms2.svg.png" decoding="async" width="250" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Vector_norms2.svg/375px-Vector_norms2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Vector_norms2.svg/500px-Vector_norms2.svg.png 2x" data-file-width="169" data-file-height="176" /></a><figcaption><a href="/wiki/Unit_ball" class="mw-redirect" title="Unit ball">Unit "spheres"</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb97520ae70482ae41b49980ec140d871cb8243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.057ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{2}}"> consist of plane vectors of norm 1. Depicted are the unit spheres in different <a href="/wiki/Lp_norm" class="mw-redirect" title="Lp norm"><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}">-norms</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1,2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1,2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e5afeb098d99e8393940f12eb4acb807507ab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.363ex; height:2.509ex;" alt="{\displaystyle p=1,2,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6ccaf28d90bcfce739aca4c5ff119a60e08031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:1.676ex;" alt="{\displaystyle \infty .}"> The bigger diamond depicts points of 1-norm equal to 2.</figcaption></figure> A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> has a limit; such a vector space is called <a href="/wiki/Completeness_(topology)" class="mw-redirect" title="Completeness (topology)">complete</a>. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971caee396752d8bf56711f55d2c3b1207d4a236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.299ex; height:2.843ex;" alt="{\displaystyle [0,1],}"> equipped with the <a href="/wiki/Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">topology of uniform convergence</a> is not complete because any continuous function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"> can be uniformly approximated by a sequence of polynomials, by the <a href="/wiki/Weierstrass_approximation_theorem" class="mw-redirect" title="Weierstrass approximation theorem">Weierstrass approximation theorem</a>.<a href="#cite_note-75">[69]</a> In contrast, the space of all continuous functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"> with the same topology is complete.<a href="#cite_note-76">[70]</a> A norm gives rise to a topology by defining that a sequence of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05afff3073235a1f9b61144677764d1e3bda26f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.629ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{n}}"> converges to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> if and only if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }|\mathbf {v} _{n}-\mathbf {v} |=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }|\mathbf {v} _{n}-\mathbf {v} |=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0483d6124c00638b669967171f5883444ea222" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.742ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }|\mathbf {v} _{n}-\mathbf {v} |=0.}"> Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.<a href="#cite_note-FOOTNOTEChoquet1966Proposition_III.7.2-77">[71]</a> The image at the right shows the equivalence of the <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}">-norm and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }">-norm on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{2}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{2}:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b5ce0ea8c16d0902300c6a7081bd5aadef39f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.349ex; height:2.676ex;" alt="{\displaystyle \mathbf {R} ^{2}:}"> as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data. From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functionals</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to W,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\to W,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52495bce078acc6cd5620da0a6da461fb6282f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.483ex; height:2.509ex;" alt="{\displaystyle V\to W,}"> maps between topological vector spaces are required to be continuous.<a href="#cite_note-FOOTNOTETreves1967p._34–36-78">[72]</a> In particular, the (topological) dual space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}"> consists of continuous functionals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\to \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30534e02fed26a26f80c4b15146efd1d141038a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.405ex; height:2.176ex;" alt="{\displaystyle V\to \mathbf {R} }"> (or to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }">). The fundamental <a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a> is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.<a href="#cite_note-FOOTNOTELang1983Cor._4.1.2,_p._69-79">[73]</a> <div class="mw-heading mw-heading4"><h4 id="Banach_spaces">Banach spaces</h4>[<a href="/w/index.php?title=Vector_space&action=edit&section=21" title="Edit section: Banach spaces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Banach_space" title="Banach space">Banach space</a></div> <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>, introduced by <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a>, are complete normed vector spaces.<a href="#cite_note-FOOTNOTETreves1967ch._11-80">[74]</a> A first example is <a href="/wiki/Lp_space" title="Lp space">the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{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 \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"></a> consisting of infinite vectors with real entries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\left(x_{1},x_{2},\ldots ,x_{n},\ldots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\left(x_{1},x_{2},\ldots ,x_{n},\ldots \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e0cca1cbc4d8b2503004b51ccc5f8c92edae1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.604ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =\left(x_{1},x_{2},\ldots ,x_{n},\ldots \right)}"> whose <a href="/wiki/P-norm" class="mw-redirect" title="P-norm"><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}">-norm</a> <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"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</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/4ec168a68b91d210bf5f71bc52715b0843c680e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.662ex; height:2.843ex;" alt="{\displaystyle (1\leq p\leq \infty )}"> given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} \|_{\infty }:=\sup _{i}|x_{i}|\qquad {\text{ for }}p=\infty ,{\text{ and }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|_{\infty }:=\sup _{i}|x_{i}|\qquad {\text{ for }}p=\infty ,{\text{ and }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee6900611e2c2424445573d079d9260b8c742f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.795ex; height:4.509ex;" alt="{\displaystyle \|\mathbf {x} \|_{\infty }:=\sup _{i}|x_{i}|\qquad {\text{ for }}p=\infty ,{\text{ and }}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i}|x_{i}|^{p}\right)^{\frac {1}{p}}\qquad {\text{ for }}p<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </msup> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i}|x_{i}|^{p}\right)^{\frac {1}{p}}\qquad {\text{ for }}p<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835dd66a07fc73c57859c8fe378dbb7354e94a0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.017ex; height:8.676ex;" alt="{\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i}|x_{i}|^{p}\right)^{\frac {1}{p}}\qquad {\text{ for }}p<\infty .}"> The topologies on the infinite-dimensional space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{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 \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> are inequivalent for different <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88532f4eab1d4cef71ef96c0f8c98cac36fd9257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p.}"> For example, the sequence of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{n}=\left(2^{-n},2^{-n},\ldots ,2^{-n},0,0,\ldots \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{n}=\left(2^{-n},2^{-n},\ldots ,2^{-n},0,0,\ldots \right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f15644e145728030235434d814c4f05d41c8fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.232ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{n}=\left(2^{-n},2^{-n},\ldots ,2^{-n},0,0,\ldots \right),}"> in which the first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"> components are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0414b5d9e81c2eb5bd85a6ca4af24b69d5336dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.659ex; height:2.509ex;" alt="{\displaystyle 2^{-n}}"> and the following ones are <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,}"> converges to the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf616a43b37b2de8dae9af2cf7ec5828055c51f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.328ex; height:2.009ex;" alt="{\displaystyle p=\infty ,}"> but does not for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=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=1:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e80f2a27ec1b538e50d813ff2aac9e5bd58a4331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.812ex; height:2.509ex;" alt="{\displaystyle p=1:}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} _{n}\|_{\infty }=\sup(2^{-n},0)=2^{-n}\to 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} _{n}\|_{\infty }=\sup(2^{-n},0)=2^{-n}\to 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca820b1d23067920b7068c104272362b13cde6a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.276ex; height:3.009ex;" alt="{\displaystyle \|\mathbf {x} _{n}\|_{\infty }=\sup(2^{-n},0)=2^{-n}\to 0,}"> but <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} _{n}\|_{1}=\sum _{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </munderover> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} _{n}\|_{1}=\sum _{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae8080b988f188916c4ee10b09e50cf0077fc89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.234ex; height:7.343ex;" alt="{\displaystyle \|\mathbf {x} _{n}\|_{1}=\sum _{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.}"> More generally than sequences of real numbers, functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\Omega \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi mathvariant="normal">Ω</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:\Omega \to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a477b266a83071d2a3bddfb5e5af1e3e105b1823" 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:\Omega \to \mathbb {R} }"> are endowed with a norm that replaces the above sum by the <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,{d\mu (x)}\right)^{\frac {1}{p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,{d\mu (x)}\right)^{\frac {1}{p}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c2bc71417915bc933a17461d3ae2cb72900f18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.228ex; height:7.176ex;" alt="{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,{d\mu (x)}\right)^{\frac {1}{p}}.}"> The space of <a href="/wiki/Integrable_function" class="mw-redirect" title="Integrable function">integrable functions</a> on a given <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 \Omega }"> (for example an interval) satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}<\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985aa50e1f0db7d175de33078041cca685111473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.732ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}<\infty ,}"> and equipped with this norm are called <a href="/wiki/Lp_space" title="Lp space">Lebesgue spaces</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\;\!p}(\Omega ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\;\!p}(\Omega ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee62bf7559ce803259cc0bff28486c8fa7ecd61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.034ex; height:2.843ex;" alt="{\displaystyle L^{\;\!p}(\Omega ).}"><a href="#cite_note-81">[nb 7]</a> These spaces are complete.<a href="#cite_note-FOOTNOTETreves1967Theorem_11.2,_p._102-82">[75]</a> (If one uses the <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a> instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.<a href="#cite_note-83">[nb 8]</a>) Concretely this means that for any sequence of Lebesgue-integrable functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},f_{2},\ldots ,f_{n},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1},f_{2},\ldots ,f_{n},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9308903d6794c2e017f99ed5f3719c87cde17e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.714ex; height:2.509ex;" alt="{\displaystyle f_{1},f_{2},\ldots ,f_{n},\ldots }"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f_{n}\|_{p}<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f_{n}\|_{p}<\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5492677d1c54ef2d39eb57059acbf1588c171f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.811ex; height:3.009ex;" alt="{\displaystyle \|f_{n}\|_{p}<\infty ,}"> satisfying the condition <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{k,\ n\to \infty }\int _{\Omega }\left|f_{k}(x)-f_{n}(x)\right|^{p}\,{d\mu (x)}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <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> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k,\ n\to \infty }\int _{\Omega }\left|f_{k}(x)-f_{n}(x)\right|^{p}\,{d\mu (x)}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9eece866e9462537619272b3fd0349a04835f58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.114ex; height:5.676ex;" alt="{\displaystyle \lim _{k,\ n\to \infty }\int _{\Omega }\left|f_{k}(x)-f_{n}(x)\right|^{p}\,{d\mu (x)}=0}"> there exists a function <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)}"> belonging to the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\;\!p}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\;\!p}(\Omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5a307f1bc9608c0233ef940dfeb0b4b9a3bef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle L^{\;\!p}(\Omega )}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{k\to \infty }\int _{\Omega }\left|f(x)-f_{k}(x)\right|^{p}\,{d\mu (x)}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <msup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }\int _{\Omega }\left|f(x)-f_{k}(x)\right|^{p}\,{d\mu (x)}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed839d6a155ecea58d76d1a88a96454d47e00dba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.657ex; height:5.676ex;" alt="{\displaystyle \lim _{k\to \infty }\int _{\Omega }\left|f(x)-f_{k}(x)\right|^{p}\,{d\mu (x)}=0.}"> Imposing boundedness conditions not only on the function, but also on its <a href="/wiki/Derivative" title="Derivative">derivatives</a> leads to <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a>.<a href="#cite_note-FOOTNOTEEvans1998ch._5-84">[76]</a> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading4"><h4 id="Hilbert_spaces">Hilbert spaces</h4>[<a href="/w/index.php?title=Vector_space&action=edit&section=22" title="Edit section: Hilbert spaces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Periodic_identity_function.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Periodic_identity_function.gif/400px-Periodic_identity_function.gif" decoding="async" width="400" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e8/Periodic_identity_function.gif 1.5x" data-file-width="491" data-file-height="126" /></a><figcaption>The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).</figcaption></figure> Complete inner product spaces are known as Hilbert spaces, in honor of <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>.<a href="#cite_note-FOOTNOTETreves1967ch._12-85">[77]</a> The Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Omega ),}"> <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> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\Omega ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd00f5ffc32ce32b622d28467b3796998859ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.771ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Omega ),}"> with inner product given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f\ ,\ g\rangle =\int _{\Omega }f(x){\overline {g(x)}}\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f\ ,\ g\rangle =\int _{\Omega }f(x){\overline {g(x)}}\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdcbee430122d5256abe43e80315dccef1cd5806" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.963ex; height:5.676ex;" alt="{\displaystyle \langle f\ ,\ g\rangle =\int _{\Omega }f(x){\overline {g(x)}}\,dx,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {g(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {g(x)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f968b7527fcc59e6aac917588d3d4c81b915828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.37ex; height:3.676ex;" alt="{\displaystyle {\overline {g(x)}}}"> denotes the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c36278728763484856f51a229bf8ce5e592e61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.902ex; height:2.843ex;" alt="{\displaystyle g(x),}"><a href="#cite_note-FOOTNOTEDenneryKrzywicki1996p.190-86">[78]</a><a href="#cite_note-87">[nb 9]</a> is a key case. By definition, in a Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions <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}}"> with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of the <a href="/wiki/Taylor_approximation" class="mw-redirect" title="Taylor approximation">Taylor approximation</a>, established an approximation of <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable functions</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}"> by polynomials.<a href="#cite_note-FOOTNOTELang1993Th._XIII.6,_p._349-88">[79]</a> By the <a href="/wiki/Stone%E2%80%93Weierstrass_theorem" title="Stone–Weierstrass theorem">Stone–Weierstrass theorem</a>, every continuous function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> can be approximated as closely as desired by a polynomial.<a href="#cite_note-FOOTNOTELang1993Th._III.1.1-89">[80]</a> A similar approximation technique by <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> is commonly called <a href="/wiki/Fourier_expansion" class="mw-redirect" title="Fourier expansion">Fourier expansion</a>, and is much applied in engineering. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef601e1519093ba6c2944b945882c119f990e704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle H,}"> in the sense that the <a href="/wiki/Closure_(topology)" title="Closure (topology)">closure</a> of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis 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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef601e1519093ba6c2944b945882c119f990e704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle H,}"> its cardinality is known as the <a href="/wiki/Hilbert_space_dimension" class="mw-redirect" title="Hilbert space dimension">Hilbert space dimension</a>.<a href="#cite_note-90">[nb 10]</a> Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the <a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a>, it enables one to construct a <a href="/wiki/Orthogonal_basis" title="Orthogonal basis">basis of orthogonal vectors</a>.<a href="#cite_note-FOOTNOTEChoquet1966Lemma_III.16.11-91">[81]</a> Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. The solutions to various <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations, and frequently solutions with particular physical properties are used as basis functions, often orthogonal.<a href="#cite_note-FOOTNOTEKreyszig1999Chapter_11-92">[82]</a> As an example from physics, the time-dependent <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> describes the change of physical properties in time by means of a <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a>, whose solutions are called <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunctions</a>.<a href="#cite_note-FOOTNOTEGriffiths1995Chapter_1-93">[83]</a> Definite values for physical properties such as energy, or momentum, correspond to <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of a certain (linear) <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> and the associated wavefunctions are called <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstates</a>. The <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> decomposes a linear <a href="/wiki/Compact_operator" title="Compact operator">compact operator</a> acting on functions in terms of these eigenfunctions and their eigenvalues.<a href="#cite_note-FOOTNOTELang1993ch._XVII.3-94">[84]</a> <div class="mw-heading mw-heading3"><h3 id="Algebras_over_fields">Algebras over fields</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=23" title="Edit section: Algebras over fields">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/Algebra_over_a_field" title="Algebra over a field">Algebra over a field</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rectangular_hyperbola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Rectangular_hyperbola.svg/250px-Rectangular_hyperbola.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Rectangular_hyperbola.svg/375px-Rectangular_hyperbola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Rectangular_hyperbola.svg/500px-Rectangular_hyperbola.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>A <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>, given by the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot y=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot y=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feae717f3cdb61fff0f74fb567b4cb24bf2825df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.072ex; height:2.509ex;" alt="{\displaystyle x\cdot y=1.}"> The <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a> of functions on this hyperbola is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} [x,y]/(x\cdot y-1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} [x,y]/(x\cdot y-1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba7424ec2e6bf0fc108cb223ae2d6209c67b44d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.602ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} [x,y]/(x\cdot y-1),}"> an infinite-dimensional vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a067bb21dcf0642bdce48f05a55e218efab3b85e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.65ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} .}"></figcaption></figure> General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional <a href="/wiki/Bilinear_operator" class="mw-redirect" title="Bilinear operator">bilinear operator</a> defining the multiplication of two vectors is an algebra over a field (or F-algebra if the field F is specified).<a href="#cite_note-FOOTNOTELang2002ch._III.1,_p._121-95">[85]</a> For example, the set of all <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b827c545ca1487214f0c498131228ef87718ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.908ex; height:2.843ex;" alt="{\displaystyle p(t)}"> forms an algebra known as the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a>: using that the sum of two polynomials is a polynomial, they form a vector space; they form an algebra since the product of two polynomials is again a polynomial. Rings of polynomials (in several variables) and their <a href="/wiki/Quotient_ring" title="Quotient ring">quotients</a> form the basis of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, because they are <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">rings of functions of algebraic geometric objects</a>.<a href="#cite_note-FOOTNOTEEisenbud1995ch._1.6-96">[86]</a> Another crucial example are <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a>, which are neither commutative nor associative, but the failure to be so is limited by the constraints (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7bd6292c6023626c6358bfd3943a031b27d663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.813ex; height:2.843ex;" alt="{\displaystyle [x,y]}"> denotes the product 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}"> 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}">): <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,y]=-[y,x]}"> <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>−</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y]=-[y,x]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70fcda86c14de45e211c3a9a889845038bb7348" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.532ex; height:2.843ex;" alt="{\displaystyle [x,y]=-[y,x]}"> (<a href="/wiki/Anticommutativity" class="mw-redirect" title="Anticommutativity">anticommutativity</a>), and</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>z</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23655a62f2a7cc545f121d9bcc30fe2c56731457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.627ex; height:2.843ex;" alt="{\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}"> (<a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a>).<a href="#cite_note-FOOTNOTEVaradarajan1974-97">[87]</a></li></ul> Examples include the vector space of <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}">-by-<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}"> matrices, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,y]=xy-yx,}"> <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> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mi>y</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y]=xy-yx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9d7bc98d1738f549c0420244c08c57decc66b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.369ex; height:2.843ex;" alt="{\displaystyle [x,y]=xy-yx,}"> the <a href="/wiki/Commutator" title="Commutator">commutator</a> of two matrices, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dabc730fd2ccb4bed18ccff4934bb48468027bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.704ex; height:3.009ex;" alt="{\displaystyle \mathbf {R} ^{3},}"> endowed with the <a href="/wiki/Cross_product" title="Cross product">cross product</a>. The <a href="/wiki/Tensor_algebra" title="Tensor algebra">tensor algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {T} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">T</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {T} (V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c18b68545f38e49986804a0e9facad8a7db9ea29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.275ex; height:2.843ex;" alt="{\displaystyle \operatorname {T} (V)}"> is a formal way of adding products to any vector space <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}"> to obtain an algebra.<a href="#cite_note-FOOTNOTELang2002ch._XVI.7-98">[88]</a> As a vector space, it is spanned by symbols, called simple <a href="/wiki/Tensor" title="Tensor">tensors</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊗</mo> <mo>⋯</mo> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7f6e265a3024dcfc8a771a92f260398aba680e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.451ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{n},}"> where the <a href="/wiki/Rank_of_a_tensor" class="mw-redirect" title="Rank of a tensor">degree</a> <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}"> varies. The multiplication is given by concatenating such symbols, imposing the <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive law</a> under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced in the above section on <a href="#Tensor_product">tensor products</a>. In general, there are no relations between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f733cf6f632a3c41f6cce290f8b45d87eeb759e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.771ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{2}\otimes \mathbf {v} _{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{2}\otimes \mathbf {v} _{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b779275d5e22fe962bc90e03258394d2ae321f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.418ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} _{2}\otimes \mathbf {v} _{1}.}"> Forcing two such elements to be equal leads to the <a href="/wiki/Symmetric_algebra" title="Symmetric algebra">symmetric algebra</a>, whereas forcing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}=-\mathbf {v} _{2}\otimes \mathbf {v} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}=-\mathbf {v} _{2}\otimes \mathbf {v} _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e33f11b00aaa607716f42fda09c86cdfc2863918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.448ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}=-\mathbf {v} _{2}\otimes \mathbf {v} _{1}}"> yields the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a>.<a href="#cite_note-FOOTNOTELang2002ch._XVI.8-99">[89]</a> <div class="mw-heading mw-heading2"><h2 id="Related_structures">Related structures</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=24" title="Edit section: Related structures">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Vector_bundles">Vector bundles</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=25" title="Edit section: Vector bundles">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/Vector_bundle" title="Vector bundle">Vector bundle</a> and <a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent bundle</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mobius_strip_illus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Mobius_strip_illus.svg/249px-Mobius_strip_illus.svg.png" decoding="async" width="249" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Mobius_strip_illus.svg/374px-Mobius_strip_illus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Mobius_strip_illus.svg/498px-Mobius_strip_illus.svg.png 2x" data-file-width="300" data-file-height="280" /></a><figcaption>A Möbius strip. Locally, it <a href="/wiki/Homeomorphism" title="Homeomorphism">looks like</a> U × R.</figcaption></figure> A vector bundle is a family of vector spaces parametrized continuously by a <a href="/wiki/Topological_space" title="Topological space">topological space</a> X.<a href="#cite_note-FOOTNOTESpivak1999ch._3-100">[90]</a> More precisely, a vector bundle over X is a topological space E equipped with a continuous map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbf8de3c25d0905f50abaf4f9374daa0a4bd796" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}"> such that for every x in X, the <a href="/wiki/Fiber_(mathematics)" title="Fiber (mathematics)">fiber</a> π−1(x) is a vector space. The case dim V = 1 is called a <a href="/wiki/Line_bundle" title="Line bundle">line bundle</a>. For any vector space V, the projection X × V → X makes the product X × V into a <a href="/wiki/Trivial_bundle" class="mw-redirect" title="Trivial bundle">"trivial" vector bundle</a>. Vector bundles over X are required to be <a href="/wiki/Locally" class="mw-redirect" title="Locally">locally</a> a product of X and some (fixed) vector space V: for every x in X, there is a <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> U of x such that the restriction of π to π−1(U) is isomorphic<a href="#cite_note-101">[nb 11]</a> to the trivial bundle U × V → U. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space X) be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle X × V). For example, the <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> can be seen as a line bundle over the circle S1 (by <a href="/wiki/Homeomorphism#Examples" title="Homeomorphism">identifying open intervals with the real line</a>). It is, however, different from the <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a> S1 × R, because the latter is <a href="/wiki/Orientable_manifold" class="mw-redirect" title="Orientable manifold">orientable</a> whereas the former is not.<a href="#cite_note-FOOTNOTEKreyszig1991§34,_p._108-102">[91]</a> Properties of certain vector bundles provide information about the underlying topological space. For example, the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> consists of the collection of <a href="/wiki/Tangent_space" title="Tangent space">tangent spaces</a> parametrized by the points of a differentiable manifold. The tangent bundle of the circle S1 is globally isomorphic to S1 × R, since there is a global nonzero <a href="/wiki/Vector_field" title="Vector field">vector field</a> on S1.<a href="#cite_note-103">[nb 12]</a> In contrast, by the <a href="/wiki/Hairy_ball_theorem" title="Hairy ball theorem">hairy ball theorem</a>, there is no (tangent) vector field on the <a href="/wiki/2-sphere" class="mw-redirect" title="2-sphere">2-sphere</a> S2 which is everywhere nonzero.<a href="#cite_note-FOOTNOTEEisenbergGuy1979-104">[92]</a> <a href="/wiki/K-theory" title="K-theory">K-theory</a> studies the isomorphism classes of all vector bundles over some topological space.<a href="#cite_note-FOOTNOTEAtiyah1989-105">[93]</a> In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real <a href="/wiki/Division_algebra" title="Division algebra">division algebras</a>: R, C, the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> H and the <a href="/wiki/Octonion" title="Octonion">octonions</a> O. The <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the <a href="/wiki/Cotangent_space" title="Cotangent space">cotangent space</a>. <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Sections</a> of that bundle are known as <a href="/wiki/Differential_form" title="Differential form">differential one-forms</a>. <div class="mw-heading mw-heading3"><h3 id="Modules">Modules</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=26" title="Edit section: Modules">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></div> Modules are to <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> what vector spaces are to fields: the same axioms, applied to a ring R instead of a field F, yield modules.<a href="#cite_note-FOOTNOTEArtin1991ch._12-106">[94]</a> The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverses</a>. For example, modules need not have bases, as the Z-module (that is, <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>) <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Z/2Z</a> shows; those modules that do (including all vector spaces) are known as <a href="/wiki/Free_module" title="Free module">free modules</a>. Nevertheless, a vector space can be compactly defined as a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> over a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> which is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, with the elements being called vectors. Some authors use the term vector space to mean modules over a <a href="/wiki/Division_ring" title="Division ring">division ring</a>.<a href="#cite_note-FOOTNOTEGrillet2007-107">[95]</a> The algebro-geometric interpretation of commutative rings via their <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum</a> allows the development of concepts such as <a href="/wiki/Locally_free_module" class="mw-redirect" title="Locally free module">locally free modules</a>, the algebraic counterpart to vector bundles. <div class="mw-heading mw-heading3"><h3 id="Affine_and_projective_spaces">Affine and projective spaces</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=27" title="Edit section: Affine and projective spaces">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/Affine_space" title="Affine space">Affine space</a> and <a href="/wiki/Projective_space" title="Projective space">Projective space</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Affine_subspace.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Affine_subspace.svg/200px-Affine_subspace.svg.png" decoding="async" width="200" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Affine_subspace.svg/300px-Affine_subspace.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Affine_subspace.svg/400px-Affine_subspace.svg.png 2x" data-file-width="216" data-file-height="166" /></a><figcaption>An <a href="/wiki/Affine_space" title="Affine space">affine plane</a> (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).</figcaption></figure> Roughly, affine spaces are vector spaces whose origins are not specified.<a href="#cite_note-FOOTNOTEMeyer2000Example_5.13.5,_p._436-108">[96]</a> More precisely, an affine space is a set with a <a href="/wiki/Transitive_group_action" class="mw-redirect" title="Transitive group action">free transitive</a> vector space <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a>. In particular, a vector space is an affine space over itself, by the map <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\times V\to W,\;(\mathbf {v} ,\mathbf {a} )\mapsto \mathbf {a} +\mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>×</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\times V\to W,\;(\mathbf {v} ,\mathbf {a} )\mapsto \mathbf {a} +\mathbf {v} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef057e61ae7876a1608903d32ac488ca66098f30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.509ex; height:2.843ex;" alt="{\displaystyle V\times V\to W,\;(\mathbf {v} ,\mathbf {a} )\mapsto \mathbf {a} +\mathbf {v} .}"> If W is a vector space, then an affine subspace is a subset of W obtained by translating a linear subspace V by a fixed vector x ∈ W; this space is denoted by x + V (it is a <a href="/wiki/Coset" title="Coset">coset</a> of V in W) and consists of all vectors of the form x + v for v ∈ V. An important example is the space of solutions of a system of inhomogeneous linear equations <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {v} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} =\mathbf {b} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daadc740034404157c9a8aeccc90455db9c8bedd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {v} =\mathbf {b} }"> generalizing the homogeneous case discussed in the <a href="#equation3">above section</a> on linear equations, which can be found by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =\mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cdf97ca5b695e4defc6395e07c3148e7e7d1af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.92ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} =\mathbf {0} }"> in this equation.<a href="#cite_note-FOOTNOTEMeyer2000Exercise_5.13.15–17,_p._442-109">[97]</a> The space of solutions is the affine subspace x + V where x is a particular solution of the equation, and V is the space of solutions of the homogeneous equation (the <a href="/wiki/Nullspace" class="mw-redirect" title="Nullspace">nullspace</a> of A). The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> lines intersecting at infinity.<a href="#cite_note-FOOTNOTECoxeter1987-110">[98]</a> <a href="/wiki/Grassmannian_manifold" class="mw-redirect" title="Grassmannian manifold">Grassmannians</a> and <a href="/wiki/Flag_manifold" class="mw-redirect" title="Flag manifold">flag manifolds</a> generalize this by parametrizing linear subspaces of fixed dimension k and <a href="/wiki/Flag_(linear_algebra)" title="Flag (linear algebra)">flags</a> of subspaces, respectively. <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=28" 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 reflist-columns references-column-width reflist-columns-3"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> It is also common, especially in physics, to denote vectors with an arrow on top: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5548bdd4889d2a7931f2478b38dded1fa898eff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.822ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}.}"> It is also common, especially in higher mathematics, to not use any typographical method for distinguishing vectors from other mathematical objects. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> Scalar multiplication is not to be confused with the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a>, which is an additional operation on some specific vector spaces, called <a href="/wiki/Inner_product_space" title="Inner product space">inner product spaces</a>. Scalar multiplication is the multiplication of a vector by a scalar that produces a vector, while the scalar product is a multiplication of two vectors that produces a scalar. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> This axiom is not an <a href="/wiki/Associative_property" title="Associative property">associative property</a>, since it refers to two different operations, scalar multiplication and field multiplication. So, it is independent from the associativity of field multiplication, which is assumed by field axioms. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> This is typically the case when a vector space is also considered as an <a href="/wiki/Affine_space" title="Affine space">affine space</a>. In this case, a linear subspace contains the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a>, while an affine subspace does not necessarily contain it. </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> Some authors, such as <a href="#CITEREFRoman2005">Roman (2005)</a>, choose to start with this <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> and derive the concrete shape of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V/W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V/W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74f0444abe38d7c9392ba2cfd13031b0f3d39cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.843ex;" alt="{\displaystyle V/W}"> from this. </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> This requirement implies that the topology gives rise to a <a href="/wiki/Uniform_structure" class="mw-redirect" title="Uniform structure">uniform structure</a>, <a href="#CITEREFBourbaki1989">Bourbaki (1989)</a>, loc = ch. II. </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> The <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d3de762058656808721fc899c4b223914c6c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.721ex; height:3.009ex;" alt="{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}"> is provided by the <a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a>. For technical reasons, in the context of functions one has to identify functions that agree <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a> to get a norm, and not only a <a href="/wiki/Seminorm" title="Seminorm">seminorm</a>. </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> "Many functions in <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}}"> of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in 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}}"> norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", <a href="#CITEREFDudley1989">Dudley (1989)</a>, §5.3, p. 125. </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\neq 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≠</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\neq 2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90cba96ef7df9709447abd7664755985c17d00ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.167ex; height:2.676ex;" alt="{\displaystyle p\neq 2,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega )}"> <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> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1ecf636f78aa6950ffe35ed4723bea9d8ca331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega )}"> is not a Hilbert space. </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> A basis of a Hilbert space is not the same thing as a basis of a linear algebra. For distinction, a linear algebra basis for a Hilbert space is called a <a href="/wiki/Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel basis</a>. </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> That is, there is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> from π−1(U) to V × U which restricts to linear isomorphisms between fibers. </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> A line bundle, such as the tangent bundle of S1 is trivial if and only if there is a <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> that vanishes nowhere, see <a href="#CITEREFHusemoller1994">Husemoller (1994)</a>, Corollary 8.3. The sections of the tangent bundle are just <a href="/wiki/Vector_field" title="Vector field">vector fields</a>. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=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 reflist-columns references-column-width" style="column-width: 20em;"> <ol class="references"> <li id="cite_note-FOOTNOTELang2002-2"><a href="#cite_ref-FOOTNOTELang2002_2-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>. </li> <li id="cite_note-FOOTNOTEBrown199186-3"><a href="#cite_ref-FOOTNOTEBrown199186_3-0">^</a> <a href="#CITEREFBrown1991">Brown 1991</a>, p. 86. </li> <li id="cite_note-FOOTNOTERoman2005ch._1,_p._27-5"><a href="#cite_ref-FOOTNOTERoman2005ch._1,_p._27_5-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 1, p. 27. </li> <li id="cite_note-FOOTNOTEBrown199187-7"><a href="#cite_ref-FOOTNOTEBrown199187_7-0">^</a> <a href="#CITEREFBrown1991">Brown 1991</a>, p. 87. </li> <li id="cite_note-FOOTNOTESpringer2000[httpsbooksgooglecombooksidCes-AAAAQBAJpgPA185_185]Brown199186-8"><a href="#cite_ref-FOOTNOTESpringer2000[httpsbooksgooglecombooksidCes-AAAAQBAJpgPA185_185]Brown199186_8-0">^</a> <a href="#CITEREFSpringer2000">Springer 2000</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ces-AAAAQBAJ&pg=PA185">185</a>; <a href="#CITEREFBrown1991">Brown 1991</a>, p. 86. </li> <li id="cite_note-FOOTNOTEAtiyahMacdonald196917-9"><a href="#cite_ref-FOOTNOTEAtiyahMacdonald196917_9-0">^</a> <a href="#CITEREFAtiyahMacdonald1969">Atiyah & Macdonald 1969</a>, p. 17. </li> <li id="cite_note-FOOTNOTEBourbaki1998§1.1,_Definition_2-10"><a href="#cite_ref-FOOTNOTEBourbaki1998§1.1,_Definition_2_10-0">^</a> <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, §1.1, Definition 2. </li> <li id="cite_note-FOOTNOTEBrown199194-11"><a href="#cite_ref-FOOTNOTEBrown199194_11-0">^</a> <a href="#CITEREFBrown1991">Brown 1991</a>, p. 94. </li> <li id="cite_note-FOOTNOTEBrown199199–101-12"><a href="#cite_ref-FOOTNOTEBrown199199–101_12-0">^</a> <a href="#CITEREFBrown1991">Brown 1991</a>, pp. 99–101. </li> <li id="cite_note-FOOTNOTEBrown199192-13"><a href="#cite_ref-FOOTNOTEBrown199192_13-0">^</a> <a href="#CITEREFBrown1991">Brown 1991</a>, p. 92. </li> <li id="cite_note-FOOTNOTEStollWong1968[httpsbooksgooglecombooksidgLbiBQAAQBAJpgPA14_14]-14">^ <a href="#cite_ref-FOOTNOTEStollWong1968[httpsbooksgooglecombooksidgLbiBQAAQBAJpgPA14_14]_14-0">a</a> <a href="#cite_ref-FOOTNOTEStollWong1968[httpsbooksgooglecombooksidgLbiBQAAQBAJpgPA14_14]_14-1">b</a> <a href="#CITEREFStollWong1968">Stoll & Wong 1968</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gLbiBQAAQBAJ&pg=PA14">14</a>. </li> <li id="cite_note-FOOTNOTERoman200541–42-15"><a href="#cite_ref-FOOTNOTERoman200541–42_15-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, pp. 41–42. </li> <li id="cite_note-FOOTNOTELang198710–11AntonRorres2010[httpsbooksgooglecombooksid1PJ-WHepeBsCpgPA212_212]-16"><a href="#cite_ref-FOOTNOTELang198710–11AntonRorres2010[httpsbooksgooglecombooksid1PJ-WHepeBsCpgPA212_212]_16-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, p. 10–11; <a href="#CITEREFAntonRorres2010">Anton & Rorres 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1PJ-WHepeBsC&pg=PA212">212</a>. </li> <li id="cite_note-FOOTNOTEBlass1984-17"><a href="#cite_ref-FOOTNOTEBlass1984_17-0">^</a> <a href="#CITEREFBlass1984">Blass 1984</a>. </li> <li id="cite_note-FOOTNOTEJoshi1989[httpsbooksgooglecombooksidRM1D3mFw2u0CpgPA450_450]-18"><a href="#cite_ref-FOOTNOTEJoshi1989[httpsbooksgooglecombooksidRM1D3mFw2u0CpgPA450_450]_18-0">^</a> <a href="#CITEREFJoshi1989">Joshi 1989</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RM1D3mFw2u0C&pg=PA450">450</a>. </li> <li id="cite_note-FOOTNOTEHeil2011[httpsbooksgooglecombooksidprfuUT0Sw-ACpgPA126_126]-19"><a href="#cite_ref-FOOTNOTEHeil2011[httpsbooksgooglecombooksidprfuUT0Sw-ACpgPA126_126]_19-0">^</a> <a href="#CITEREFHeil2011">Heil 2011</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=prfuUT0Sw-AC&pg=PA126">126</a>. </li> <li id="cite_note-FOOTNOTEHalmos1948[httpsbooksgooglecombooksid1hzYCwAAQBAJpgPA12_12]-20"><a href="#cite_ref-FOOTNOTEHalmos1948[httpsbooksgooglecombooksid1hzYCwAAQBAJpgPA12_12]_20-0">^</a> <a href="#CITEREFHalmos1948">Halmos 1948</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1hzYCwAAQBAJ&pg=PA12">12</a>. </li> <li id="cite_note-FOOTNOTEBourbaki1969ch._"Algèbre_linéaire_et_algèbre_multilinéaire",_pp._78–91-21"><a href="#cite_ref-FOOTNOTEBourbaki1969ch._"Algèbre_linéaire_et_algèbre_multilinéaire",_pp._78–91_21-0">^</a> <a href="#CITEREFBourbaki1969">Bourbaki 1969</a>, ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78–91. </li> <li id="cite_note-FOOTNOTEBolzano1804-22"><a href="#cite_ref-FOOTNOTEBolzano1804_22-0">^</a> <a href="#CITEREFBolzano1804">Bolzano 1804</a>. </li> <li id="cite_note-FOOTNOTEMöbius1827-23"><a href="#cite_ref-FOOTNOTEMöbius1827_23-0">^</a> <a href="#CITEREFMöbius1827">Möbius 1827</a>. </li> <li id="cite_note-FOOTNOTEBellavitis1833-24"><a href="#cite_ref-FOOTNOTEBellavitis1833_24-0">^</a> <a href="#CITEREFBellavitis1833">Bellavitis 1833</a>. </li> <li id="cite_note-FOOTNOTEDorier1995-25"><a href="#cite_ref-FOOTNOTEDorier1995_25-0">^</a> <a href="#CITEREFDorier1995">Dorier 1995</a>. </li> <li id="cite_note-FOOTNOTEHamilton1853-26"><a href="#cite_ref-FOOTNOTEHamilton1853_26-0">^</a> <a href="#CITEREFHamilton1853">Hamilton 1853</a>. </li> <li id="cite_note-FOOTNOTEGrassmann2000-27"><a href="#cite_ref-FOOTNOTEGrassmann2000_27-0">^</a> <a href="#CITEREFGrassmann2000">Grassmann 2000</a>. </li> <li id="cite_note-FOOTNOTEPeano1888ch._IX-28"><a href="#cite_ref-FOOTNOTEPeano1888ch._IX_28-0">^</a> <a href="#CITEREFPeano1888">Peano 1888</a>, ch. IX. </li> <li id="cite_note-FOOTNOTEGuo2021-29"><a href="#cite_ref-FOOTNOTEGuo2021_29-0">^</a> <a href="#CITEREFGuo2021">Guo 2021</a>. </li> <li id="cite_note-FOOTNOTEMoore1995268–271-30"><a href="#cite_ref-FOOTNOTEMoore1995268–271_30-0">^</a> <a href="#CITEREFMoore1995">Moore 1995</a>, pp. 268–271. </li> <li id="cite_note-FOOTNOTEBanach1922-31"><a href="#cite_ref-FOOTNOTEBanach1922_31-0">^</a> <a href="#CITEREFBanach1922">Banach 1922</a>. </li> <li id="cite_note-FOOTNOTEDorier1995Moore1995-32"><a href="#cite_ref-FOOTNOTEDorier1995Moore1995_32-0">^</a> <a href="#CITEREFDorier1995">Dorier 1995</a>; <a href="#CITEREFMoore1995">Moore 1995</a>. </li> <li id="cite_note-FOOTNOTEKreyszig2020[httpsbooksgooglecombooksidw4T3DwAAQBAJpgPA355_355]-33"><a href="#cite_ref-FOOTNOTEKreyszig2020[httpsbooksgooglecombooksidw4T3DwAAQBAJpgPA355_355]_33-0">^</a> <a href="#CITEREFKreyszig2020">Kreyszig 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA355">355</a>. </li> <li id="cite_note-FOOTNOTEKreyszig2020[httpsbooksgooglecombooksidw4T3DwAAQBAJpgPA358_358&ndash;359]-34"><a href="#cite_ref-FOOTNOTEKreyszig2020[httpsbooksgooglecombooksidw4T3DwAAQBAJpgPA358_358&ndash;359]_34-0">^</a> <a href="#CITEREFKreyszig2020">Kreyszig 2020</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA358">358–359</a>. </li> <li id="cite_note-FOOTNOTEJain2001[httpsbooksgooglecombooksid-lzAee3uQtICpgPA11_11]-35"><a href="#cite_ref-FOOTNOTEJain2001[httpsbooksgooglecombooksid-lzAee3uQtICpgPA11_11]_35-0">^</a> <a href="#CITEREFJain2001">Jain 2001</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-lzAee3uQtIC&pg=PA11">11</a>. </li> <li id="cite_note-FOOTNOTELang1987ch._I.1-36"><a href="#cite_ref-FOOTNOTELang1987ch._I.1_36-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, ch. I.1. </li> <li id="cite_note-FOOTNOTELang2002ch._V.1-37"><a href="#cite_ref-FOOTNOTELang2002ch._V.1_37-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, ch. V.1. </li> <li id="cite_note-FOOTNOTELang1993ch._XII.3.,_p._335-38"><a href="#cite_ref-FOOTNOTELang1993ch._XII.3.,_p._335_38-0">^</a> <a href="#CITEREFLang1993">Lang 1993</a>, ch. XII.3., p. 335. </li> <li id="cite_note-FOOTNOTELang1987ch._VI.3.-39"><a href="#cite_ref-FOOTNOTELang1987ch._VI.3._39-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, ch. VI.3.. </li> <li id="cite_note-FOOTNOTERoman2005ch._2,_p._45-40"><a href="#cite_ref-FOOTNOTERoman2005ch._2,_p._45_40-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 2, p. 45. </li> <li id="cite_note-FOOTNOTELang1987ch._IV.4,_Corollary,_p._106-41"><a href="#cite_ref-FOOTNOTELang1987ch._IV.4,_Corollary,_p._106_41-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, ch. IV.4, Corollary, p. 106. </li> <li id="cite_note-FOOTNOTENicholson2018ch._7.3-42"><a href="#cite_ref-FOOTNOTENicholson2018ch._7.3_42-0">^</a> <a href="#CITEREFNicholson2018">Nicholson 2018</a>, ch. 7.3. </li> <li id="cite_note-FOOTNOTELang1987Example_IV.2.6-43"><a href="#cite_ref-FOOTNOTELang1987Example_IV.2.6_43-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, Example IV.2.6. </li> <li id="cite_note-FOOTNOTELang1987ch._VI.6-44"><a href="#cite_ref-FOOTNOTELang1987ch._VI.6_44-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, ch. VI.6. </li> <li id="cite_note-FOOTNOTEHalmos1974p._28,_Ex._9-45"><a href="#cite_ref-FOOTNOTEHalmos1974p._28,_Ex._9_45-0">^</a> <a href="#CITEREFHalmos1974">Halmos 1974</a>, p. 28, Ex. 9. </li> <li id="cite_note-FOOTNOTELang1987Theorem_IV.2.1,_p._95-46"><a href="#cite_ref-FOOTNOTELang1987Theorem_IV.2.1,_p._95_46-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, Theorem IV.2.1, p. 95. </li> <li id="cite_note-FOOTNOTERoman2005Th._2.5_and_2.6,_p._49-47"><a href="#cite_ref-FOOTNOTERoman2005Th._2.5_and_2.6,_p._49_47-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, Th. 2.5 and 2.6, p. 49. </li> <li id="cite_note-FOOTNOTELang1987ch._V.1-48"><a href="#cite_ref-FOOTNOTELang1987ch._V.1_48-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, ch. V.1. </li> <li id="cite_note-FOOTNOTELang1987ch._V.3.,_Corollary,_p._106-49"><a href="#cite_ref-FOOTNOTELang1987ch._V.3.,_Corollary,_p._106_49-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, ch. V.3., Corollary, p. 106. </li> <li id="cite_note-FOOTNOTELang1987Theorem_VII.9.8,_p._198-50"><a href="#cite_ref-FOOTNOTELang1987Theorem_VII.9.8,_p._198_50-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, Theorem VII.9.8, p. 198. </li> <li id="cite_note-FOOTNOTERoman2005ch._8,_p._135–156-51"><a href="#cite_ref-FOOTNOTERoman2005ch._8,_p._135–156_51-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 8, p. 135–156. </li> <li id="cite_note-FOOTNOTELang1987ch._IX.4-52"><a href="#cite_ref-FOOTNOTELang1987ch._IX.4_52-0">^</a> <a href="#CITEREFLang1987"> & Lang 1987</a>, ch. IX.4. </li> <li id="cite_note-FOOTNOTERoman2005ch._8,_p._140-53"><a href="#cite_ref-FOOTNOTERoman2005ch._8,_p._140_53-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 8, p. 140. </li> <li id="cite_note-FOOTNOTERoman2005ch._1,_p._29-54"><a href="#cite_ref-FOOTNOTERoman2005ch._1,_p._29_54-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 1, p. 29. </li> <li id="cite_note-FOOTNOTERoman2005ch._1,_p._35-56"><a href="#cite_ref-FOOTNOTERoman2005ch._1,_p._35_56-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 1, p. 35. </li> <li id="cite_note-FOOTNOTENicholson2018ch._10.4-57"><a href="#cite_ref-FOOTNOTENicholson2018ch._10.4_57-0">^</a> <a href="#CITEREFNicholson2018">Nicholson 2018</a>, ch. 10.4. </li> <li id="cite_note-FOOTNOTERoman2005ch._3,_p._64-58"><a href="#cite_ref-FOOTNOTERoman2005ch._3,_p._64_58-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 3, p. 64. </li> <li id="cite_note-FOOTNOTELang1987ch._IV.3.-60"><a href="#cite_ref-FOOTNOTELang1987ch._IV.3._60-0">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, ch. IV.3.. </li> <li id="cite_note-FOOTNOTERoman2005ch._2,_p._48-61"><a href="#cite_ref-FOOTNOTERoman2005ch._2,_p._48_61-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 2, p. 48. </li> <li id="cite_note-FOOTNOTENicholson2018ch._7.4-62"><a href="#cite_ref-FOOTNOTENicholson2018ch._7.4_62-0">^</a> <a href="#CITEREFNicholson2018">Nicholson 2018</a>, ch. 7.4. </li> <li id="cite_note-FOOTNOTEMac_Lane1998-63"><a href="#cite_ref-FOOTNOTEMac_Lane1998_63-0">^</a> <a href="#CITEREFMac_Lane1998">Mac Lane 1998</a>. </li> <li id="cite_note-FOOTNOTERoman2005ch._1,_pp._31–32-64"><a href="#cite_ref-FOOTNOTERoman2005ch._1,_pp._31–32_64-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 1, pp. 31–32. </li> <li id="cite_note-FOOTNOTELang2002ch._XVI.1-65"><a href="#cite_ref-FOOTNOTELang2002ch._XVI.1_65-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, ch. XVI.1. </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <a href="#CITEREFRoman2005">Roman (2005)</a>, Th. 14.3. See also <a href="/wiki/Yoneda_lemma" title="Yoneda lemma">Yoneda lemma</a>. </li> <li id="cite_note-FOOTNOTERudin1991p.3-67"><a href="#cite_ref-FOOTNOTERudin1991p.3_67-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p.3. </li> <li id="cite_note-FOOTNOTESchaeferWolff1999pp._204–205-68"><a href="#cite_ref-FOOTNOTESchaeferWolff1999pp._204–205_68-0">^</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, pp. 204–205. </li> <li id="cite_note-FOOTNOTEBourbaki2004ch._2,_p._48-69"><a href="#cite_ref-FOOTNOTEBourbaki2004ch._2,_p._48_69-0">^</a> <a href="#CITEREFBourbaki2004">Bourbaki 2004</a>, ch. 2, p. 48. </li> <li id="cite_note-FOOTNOTERoman2005ch._9-70"><a href="#cite_ref-FOOTNOTERoman2005ch._9_70-0">^</a> <a href="#CITEREFRoman2005">Roman 2005</a>, ch. 9. </li> <li id="cite_note-FOOTNOTENaber2003ch._1.2-71"><a href="#cite_ref-FOOTNOTENaber2003ch._1.2_71-0">^</a> <a href="#CITEREFNaber2003">Naber 2003</a>, ch. 1.2. </li> <li id="cite_note-FOOTNOTETreves1967Bourbaki1987-72"><a href="#cite_ref-FOOTNOTETreves1967Bourbaki1987_72-0">^</a> <a href="#CITEREFTreves1967">Treves 1967</a>; <a href="#CITEREFBourbaki1987">Bourbaki 1987</a>. </li> <li id="cite_note-FOOTNOTESchaeferWolff1999p._7-74"><a href="#cite_ref-FOOTNOTESchaeferWolff1999p._7_74-0">^</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, p. 7. </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <a href="#CITEREFKreyszig1989">Kreyszig 1989</a>, §4.11-5 </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <a href="#CITEREFKreyszig1989">Kreyszig 1989</a>, §1.5-5 </li> <li id="cite_note-FOOTNOTEChoquet1966Proposition_III.7.2-77"><a href="#cite_ref-FOOTNOTEChoquet1966Proposition_III.7.2_77-0">^</a> <a href="#CITEREFChoquet1966">Choquet 1966</a>, Proposition III.7.2. </li> <li id="cite_note-FOOTNOTETreves1967p._34–36-78"><a href="#cite_ref-FOOTNOTETreves1967p._34–36_78-0">^</a> <a href="#CITEREFTreves1967">Treves 1967</a>, p. 34–36. </li> <li id="cite_note-FOOTNOTELang1983Cor._4.1.2,_p._69-79"><a href="#cite_ref-FOOTNOTELang1983Cor._4.1.2,_p._69_79-0">^</a> <a href="#CITEREFLang1983">Lang 1983</a>, Cor. 4.1.2, p. 69. </li> <li id="cite_note-FOOTNOTETreves1967ch._11-80"><a href="#cite_ref-FOOTNOTETreves1967ch._11_80-0">^</a> <a href="#CITEREFTreves1967">Treves 1967</a>, ch. 11. </li> <li id="cite_note-FOOTNOTETreves1967Theorem_11.2,_p._102-82"><a href="#cite_ref-FOOTNOTETreves1967Theorem_11.2,_p._102_82-0">^</a> <a href="#CITEREFTreves1967">Treves 1967</a>, Theorem 11.2, p. 102. </li> <li id="cite_note-FOOTNOTEEvans1998ch._5-84"><a href="#cite_ref-FOOTNOTEEvans1998ch._5_84-0">^</a> <a href="#CITEREFEvans1998">Evans 1998</a>, ch. 5. </li> <li id="cite_note-FOOTNOTETreves1967ch._12-85"><a href="#cite_ref-FOOTNOTETreves1967ch._12_85-0">^</a> <a href="#CITEREFTreves1967">Treves 1967</a>, ch. 12. </li> <li id="cite_note-FOOTNOTEDenneryKrzywicki1996p.190-86"><a href="#cite_ref-FOOTNOTEDenneryKrzywicki1996p.190_86-0">^</a> <a href="#CITEREFDenneryKrzywicki1996">Dennery & Krzywicki 1996</a>, p.190. </li> <li id="cite_note-FOOTNOTELang1993Th._XIII.6,_p._349-88"><a href="#cite_ref-FOOTNOTELang1993Th._XIII.6,_p._349_88-0">^</a> <a href="#CITEREFLang1993">Lang 1993</a>, Th. XIII.6, p. 349. </li> <li id="cite_note-FOOTNOTELang1993Th._III.1.1-89"><a href="#cite_ref-FOOTNOTELang1993Th._III.1.1_89-0">^</a> <a href="#CITEREFLang1993">Lang 1993</a>, Th. III.1.1. </li> <li id="cite_note-FOOTNOTEChoquet1966Lemma_III.16.11-91"><a href="#cite_ref-FOOTNOTEChoquet1966Lemma_III.16.11_91-0">^</a> <a href="#CITEREFChoquet1966">Choquet 1966</a>, Lemma III.16.11. </li> <li id="cite_note-FOOTNOTEKreyszig1999Chapter_11-92"><a href="#cite_ref-FOOTNOTEKreyszig1999Chapter_11_92-0">^</a> <a href="#CITEREFKreyszig1999">Kreyszig 1999</a>, Chapter 11. </li> <li id="cite_note-FOOTNOTEGriffiths1995Chapter_1-93"><a href="#cite_ref-FOOTNOTEGriffiths1995Chapter_1_93-0">^</a> <a href="#CITEREFGriffiths1995">Griffiths 1995</a>, Chapter 1. </li> <li id="cite_note-FOOTNOTELang1993ch._XVII.3-94"><a href="#cite_ref-FOOTNOTELang1993ch._XVII.3_94-0">^</a> <a href="#CITEREFLang1993">Lang 1993</a>, ch. XVII.3. </li> <li id="cite_note-FOOTNOTELang2002ch._III.1,_p._121-95"><a href="#cite_ref-FOOTNOTELang2002ch._III.1,_p._121_95-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, ch. III.1, p. 121. </li> <li id="cite_note-FOOTNOTEEisenbud1995ch._1.6-96"><a href="#cite_ref-FOOTNOTEEisenbud1995ch._1.6_96-0">^</a> <a href="#CITEREFEisenbud1995">Eisenbud 1995</a>, ch. 1.6. </li> <li id="cite_note-FOOTNOTEVaradarajan1974-97"><a href="#cite_ref-FOOTNOTEVaradarajan1974_97-0">^</a> <a href="#CITEREFVaradarajan1974">Varadarajan 1974</a>. </li> <li id="cite_note-FOOTNOTELang2002ch._XVI.7-98"><a href="#cite_ref-FOOTNOTELang2002ch._XVI.7_98-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, ch. XVI.7. </li> <li id="cite_note-FOOTNOTELang2002ch._XVI.8-99"><a href="#cite_ref-FOOTNOTELang2002ch._XVI.8_99-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, ch. XVI.8. </li> <li id="cite_note-FOOTNOTESpivak1999ch._3-100"><a href="#cite_ref-FOOTNOTESpivak1999ch._3_100-0">^</a> <a href="#CITEREFSpivak1999">Spivak 1999</a>, ch. 3. </li> <li id="cite_note-FOOTNOTEKreyszig1991§34,_p._108-102"><a href="#cite_ref-FOOTNOTEKreyszig1991§34,_p._108_102-0">^</a> <a href="#CITEREFKreyszig1991">Kreyszig 1991</a>, §34, p. 108. </li> <li id="cite_note-FOOTNOTEEisenbergGuy1979-104"><a href="#cite_ref-FOOTNOTEEisenbergGuy1979_104-0">^</a> <a href="#CITEREFEisenbergGuy1979">Eisenberg & Guy 1979</a>. </li> <li id="cite_note-FOOTNOTEAtiyah1989-105"><a href="#cite_ref-FOOTNOTEAtiyah1989_105-0">^</a> <a href="#CITEREFAtiyah1989">Atiyah 1989</a>. </li> <li id="cite_note-FOOTNOTEArtin1991ch._12-106"><a href="#cite_ref-FOOTNOTEArtin1991ch._12_106-0">^</a> <a href="#CITEREFArtin1991">Artin 1991</a>, ch. 12. </li> <li id="cite_note-FOOTNOTEGrillet2007-107"><a href="#cite_ref-FOOTNOTEGrillet2007_107-0">^</a> <a href="#CITEREFGrillet2007">Grillet 2007</a>. </li> <li id="cite_note-FOOTNOTEMeyer2000Example_5.13.5,_p._436-108"><a href="#cite_ref-FOOTNOTEMeyer2000Example_5.13.5,_p._436_108-0">^</a> <a href="#CITEREFMeyer2000">Meyer 2000</a>, Example 5.13.5, p. 436. </li> <li id="cite_note-FOOTNOTEMeyer2000Exercise_5.13.15–17,_p._442-109"><a href="#cite_ref-FOOTNOTEMeyer2000Exercise_5.13.15–17,_p._442_109-0">^</a> <a href="#CITEREFMeyer2000">Meyer 2000</a>, Exercise 5.13.15–17, p. 442. </li> <li id="cite_note-FOOTNOTECoxeter1987-110"><a href="#cite_ref-FOOTNOTECoxeter1987_110-0">^</a> <a href="#CITEREFCoxeter1987">Coxeter 1987</a>. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=30" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Algebra">Algebra</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=31" title="Edit section: Algebra">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><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAntonRorres2010" class="citation cs2">Anton, Howard; Rorres, Chris (2010), Elementary Linear Algebra: Applications Version (10th ed.), John Wiley & Sons</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra%3A+Applications+Version&rft.edition=10th&rft.pub=John+Wiley+%26+Sons&rft.date=2010&rft.aulast=Anton&rft.aufirst=Howard&rft.au=Rorres%2C+Chris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin1991" class="citation cs2"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (1991), Algebra, <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-510-1" 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(1968), Linear Algebra, Academic Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.pub=Academic+Press&rft.date=1968&rft.aulast=Stoll&rft.aufirst=R.+R.&rft.au=Wong%2C+E.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Waerden1993" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, Bartel Leendert</a> (1993), Algebra (in German) (9th ed.), 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/978-3-540-56799-8" 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.place=Boston%2C+MA&rft.pub=Academic+Press&rft.date=1966&rft.aulast=Choquet&rft.aufirst=Gustave&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDenneryKrzywicki1996" class="citation cs2">Dennery, Philippe; Krzywicki, Andre (1996), Mathematics for Physicists, Courier Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-69193-0" title="Special:BookSources/978-0-486-69193-0"><bdi>978-0-486-69193-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Physicists&rft.pub=Courier+Dover+Publications&rft.date=1996&rft.isbn=978-0-486-69193-0&rft.aulast=Dennery&rft.aufirst=Philippe&rft.au=Krzywicki%2C+Andre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDudley1989" class="citation cs2">Dudley, Richard M. (1989), Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-10050-6" title="Special:BookSources/978-0-534-10050-6"><bdi>978-0-534-10050-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+analysis+and+probability&rft.place=Pacific+Grove%2C+CA&rft.series=The+Wadsworth+%26+Brooks%2FCole+Mathematics+Series&rft.pub=Wadsworth+%26+Brooks%2FCole+Advanced+Books+%26+Software&rft.date=1989&rft.isbn=978-0-534-10050-6&rft.aulast=Dudley&rft.aufirst=Richard+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunham2005" class="citation cs2">Dunham, William (2005), The Calculus Gallery, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-09565-3" title="Special:BookSources/978-0-691-09565-3"><bdi>978-0-691-09565-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Calculus+Gallery&rft.pub=Princeton+University+Press&rft.date=2005&rft.isbn=978-0-691-09565-3&rft.aulast=Dunham&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans1998" class="citation cs2">Evans, Lawrence C. (1998), Partial differential equations, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0772-9" title="Special:BookSources/978-0-8218-0772-9"><bdi>978-0-8218-0772-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+differential+equations&rft.place=Providence%2C+R.I.&rft.pub=American+Mathematical+Society&rft.date=1998&rft.isbn=978-0-8218-0772-9&rft.aulast=Evans&rft.aufirst=Lawrence+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland1992" class="citation cs2">Folland, Gerald B. (1992), Fourier Analysis and Its Applications, Brooks-Cole, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-17094-3" title="Special:BookSources/978-0-534-17094-3"><bdi>978-0-534-17094-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Analysis+and+Its+Applications&rft.pub=Brooks-Cole&rft.date=1992&rft.isbn=978-0-534-17094-3&rft.aulast=Folland&rft.aufirst=Gerald+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGasquetWitomski1999" class="citation cs2">Gasquet, Claude; Witomski, Patrick (1999), Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets, Texts in Applied Mathematics, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98485-8" title="Special:BookSources/978-0-387-98485-8"><bdi>978-0-387-98485-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Analysis+and+Applications%3A+Filtering%2C+Numerical+Computation%2C+Wavelets&rft.place=New+York&rft.series=Texts+in+Applied+Mathematics&rft.pub=Springer-Verlag&rft.date=1999&rft.isbn=978-0-387-98485-8&rft.aulast=Gasquet&rft.aufirst=Claude&rft.au=Witomski%2C+Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIfeachorJervis2001" class="citation cs2">Ifeachor, Emmanuel C.; Jervis, Barrie W. (2001), Digital Signal Processing: A Practical Approach (2nd ed.), Harlow, Essex, England: Prentice-Hall (published 2002), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-59619-9" title="Special:BookSources/978-0-201-59619-9"><bdi>978-0-201-59619-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Signal+Processing%3A+A+Practical+Approach&rft.place=Harlow%2C+Essex%2C+England&rft.edition=2nd&rft.pub=Prentice-Hall&rft.date=2001&rft.isbn=978-0-201-59619-9&rft.aulast=Ifeachor&rft.aufirst=Emmanuel+C.&rft.au=Jervis%2C+Barrie+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrantz1999" class="citation cs2">Krantz, Steven G. (1999), A Panorama of Harmonic Analysis, Carus Mathematical Monographs, Washington, DC: Mathematical Association of America, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-031-2" title="Special:BookSources/978-0-88385-031-2"><bdi>978-0-88385-031-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Panorama+of+Harmonic+Analysis&rft.place=Washington%2C+DC&rft.series=Carus+Mathematical+Monographs&rft.pub=Mathematical+Association+of+America&rft.date=1999&rft.isbn=978-0-88385-031-2&rft.aulast=Krantz&rft.aufirst=Steven+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKreyszig1988" class="citation cs2"><a href="/wiki/Erwin_Kreyszig" title="Erwin Kreyszig">Kreyszig, Erwin</a> (1988), Advanced Engineering Mathematics (6th ed.), New York: John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-85824-9" title="Special:BookSources/978-0-471-85824-9"><bdi>978-0-471-85824-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Engineering+Mathematics&rft.place=New+York&rft.edition=6th&rft.pub=John+Wiley+%26+Sons&rft.date=1988&rft.isbn=978-0-471-85824-9&rft.aulast=Kreyszig&rft.aufirst=Erwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKreyszig1989" class="citation cs2"><a href="/wiki/Erwin_Kreyszig" title="Erwin Kreyszig">Kreyszig, Erwin</a> (1989), Introductory functional analysis with applications, Wiley Classics Library, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-50459-7" title="Special:BookSources/978-0-471-50459-7"><bdi>978-0-471-50459-7</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=0992618">0992618</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+functional+analysis+with+applications&rft.place=New+York&rft.series=Wiley+Classics+Library&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=978-0-471-50459-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D992618%23id-name%3DMR&rft.aulast=Kreyszig&rft.aufirst=Erwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1983" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1983), Real analysis, <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-14179-5" title="Special:BookSources/978-0-201-14179-5"><bdi>978-0-201-14179-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+analysis&rft.pub=Addison-Wesley&rft.date=1983&rft.isbn=978-0-201-14179-5&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1993" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1993), Real and functional analysis, 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/978-0-387-94001-4" title="Special:BookSources/978-0-387-94001-4"><bdi>978-0-387-94001-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+functional+analysis&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1993&rft.isbn=978-0-387-94001-4&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoomis2011" class="citation cs2">Loomis, Lynn H. (2011) [1953], An introduction to abstract harmonic analysis, Dover, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fuc1.b4250788">2027/uc1.b4250788</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-48123-4" title="Special:BookSources/978-0-486-48123-4"><bdi>978-0-486-48123-4</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/702357363">702357363</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+abstract+harmonic+analysis&rft.pub=Dover&rft.date=2011&rft_id=info%3Ahdl%2F2027%2Fuc1.b4250788&rft_id=info%3Aoclcnum%2F702357363&rft.isbn=978-0-486-48123-4&rft.aulast=Loomis&rft.aufirst=Lynn+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+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%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1991" class="citation cs2">Rudin, Walter (1991), Functional analysis (2 ed.), McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0070542368" title="Special:BookSources/0070542368"><bdi>0070542368</bdi></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.edition=2&rft.pub=McGraw-Hill&rft.date=1991&rft.isbn=0070542368&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+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. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). 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%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTreves1967" class="citation cs2"><a href="/wiki/Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Treves, François</a> (1967), Topological vector spaces, distributions and kernels, Boston, MA: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</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=Boston%2C+MA&rft.pub=Academic+Press&rft.date=1967&rft.aulast=Treves&rft.aufirst=Fran%C3%A7ois&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Historical_references">Historical references</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=33" title="Edit section: Historical references">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><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="CITEREFBanach1922" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach, Stefan</a> (1922), <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3120.pdf">"Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)"</a> (PDF), <a href="/wiki/Fundamenta_Mathematicae" title="Fundamenta Mathematicae">Fundamenta Mathematicae</a> (in French), 3: 133–181, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Ffm-3-1-133-181">10.4064/fm-3-1-133-181</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0016-2736">0016-2736</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fundamenta+Mathematicae&rft.atitle=Sur+les+op%C3%A9rations+dans+les+ensembles+abstraits+et+leur+application+aux+%C3%A9quations+int%C3%A9grales+%28On+operations+in+abstract+sets+and+their+application+to+integral+equations%29&rft.volume=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E133-%3C%2Fspan%3E181&rft.date=1922&rft_id=info%3Adoi%2F10.4064%2Ffm-3-1-133-181&rft.issn=0016-2736&rft.aulast=Banach&rft.aufirst=Stefan&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm3%2Ffm3120.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBolzano1804" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano, Bernard</a> (1804), <a rel="nofollow" class="external text" href="http://dml.cz/handle/10338.dmlcz/400338">Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry)</a> (in German)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Betrachtungen+%C3%BCber+einige+Gegenst%C3%A4nde+der+Elementargeometrie+%28Considerations+of+some+aspects+of+elementary+geometry%29&rft.date=1804&rft.aulast=Bolzano&rft.aufirst=Bernard&rft_id=http%3A%2F%2Fdml.cz%2Fhandle%2F10338.dmlcz%2F400338&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBellavitis1833" class="citation cs2"><a href="/wiki/Giusto_Bellavitis" title="Giusto Bellavitis">Bellavitis, Giuso</a> (1833), "Sopra alcune applicazioni di un nuovo 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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=1347828">1347828</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=A+general+outline+of+the+genesis+of+vector+space+theory&rft.volume=22&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E227-%3C%2Fspan%3E261&rft.date=1995&rft_id=info%3Adoi%2F10.1006%2Fhmat.1995.1024&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1347828%23id-name%3DMR&rft.aulast=Dorier&rft.aufirst=Jean-Luc&rft_id=http%3A%2F%2Farchive-ouverte.unige.ch%2Funige%3A16642&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFourier1822" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier, Jean Baptiste Joseph</a> (1822), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TDQJAAAAIAAJ">Théorie analytique de la chaleur</a> (in French), Chez Firmin Didot, père et fils</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+analytique+de+la+chaleur&rft.pub=Chez+Firmin+Didot%2C+p%C3%A8re+et+fils&rft.date=1822&rft.aulast=Fourier&rft.aufirst=Jean+Baptiste+Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTDQJAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrassmann1844" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Grassmann, Hermann</a> (1844), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bKgAAAAAMAAJ&pg=PA1">Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik</a> (in German), O. Wigand</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+Lineale+Ausdehnungslehre+-+Ein+neuer+Zweig+der+Mathematik&rft.pub=O.+Wigand&rft.date=1844&rft.aulast=Grassmann&rft.aufirst=Hermann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbKgAAAAAMAAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988">, reprint: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrassmann2000" class="citation cs2">Grassmann, Hermann (2000), Kannenberg, L.C. 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(1995), "The axiomatization of linear algebra: 1875–1940", <a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a>, 22 (3): 262–303, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.1995.1025">10.1006/hmat.1995.1025</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+axiomatization+of+linear+algebra%3A+1875%E2%80%931940&rft.volume=22&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E262-%3C%2Fspan%3E303&rft.date=1995&rft_id=info%3Adoi%2F10.1006%2Fhmat.1995.1025&rft.aulast=Moore&rft.aufirst=Gregory+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1888" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian), Turin</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calcolo+Geometrico+secondo+l%27Ausdehnungslehre+di+H.+Grassmann+preceduto+dalle+Operazioni+della+Logica+Deduttiva&rft.place=Turin&rft.date=1888&rft.aulast=Peano&rft.aufirst=Giuseppe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</li> <li>Peano, G. (1901) <a href="/wiki/Formulario_mathematico" title="Formulario mathematico">Formulario mathematico</a>: <a rel="nofollow" class="external text" href="https://archive.org/details/formulairedesmat00pean/page/194">vct axioms</a> via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Further_references">Further references</h3>[<a href="/w/index.php?title=Vector_space&action=edit&section=34" title="Edit section: Further references">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><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="CITEREFAshcroftMermin1976" class="citation cs2"><a href="/wiki/Neil_Ashcroft" title="Neil Ashcroft">Ashcroft, Neil</a>; <a href="/wiki/N._David_Mermin" title="N. David Mermin">Mermin, N. David</a> (1976), <a rel="nofollow" class="external text" href="https://archive.org/details/solidstatephysic00ashc">Solid State Physics</a>, Toronto: Thomson Learning, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-083993-1" title="Special:BookSources/978-0-03-083993-1"><bdi>978-0-03-083993-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solid+State+Physics&rft.place=Toronto&rft.pub=Thomson+Learning&rft.date=1976&rft.isbn=978-0-03-083993-1&rft.aulast=Ashcroft&rft.aufirst=Neil&rft.au=Mermin%2C+N.+David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsolidstatephysic00ashc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtiyah1989" class="citation cs2"><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah, Michael Francis</a> (1989), K-theory, Advanced Book Classics (2nd ed.), <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-09394-0" title="Special:BookSources/978-0-201-09394-0"><bdi>978-0-201-09394-0</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=1043170">1043170</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=K-theory&rft.series=Advanced+Book+Classics&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1989&rft.isbn=978-0-201-09394-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1043170%23id-name%3DMR&rft.aulast=Atiyah&rft.aufirst=Michael+Francis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtiyahMacdonald1969" class="citation cs2"><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah, Michael Francis</a>; <a href="/wiki/Ian_G._Macdonald" title="Ian G. 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class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66721-8" title="Special:BookSources/978-0-486-66721-8"><bdi>978-0-486-66721-8</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+geometry&rft.place=New+York&rft.pages=xiv%2B352&rft.pub=Dover+Publications&rft.date=1991&rft.isbn=978-0-486-66721-8&rft.aulast=Kreyszig&rft.aufirst=Erwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKreyszig1999" class="citation cs2">Kreyszig, Erwin (1999), <a rel="nofollow" class="external text" href="https://archive.org/details/advancedengineer0008krey">Advanced Engineering Mathematics</a> (8th ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" 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title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-18117-0" title="Special:BookSources/978-0-471-18117-0"><bdi>978-0-471-18117-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimization+by+vector+space+methods&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1997&rft.isbn=978-0-471-18117-0&rft.aulast=Luenberger&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_Lane1998" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1998), <a href="/wiki/Categories_for_the_Working_Mathematician" title="Categories for the Working Mathematician">Categories for the Working Mathematician</a> (2nd ed.), 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/978-0-387-98403-2" title="Special:BookSources/978-0-387-98403-2"><bdi>978-0-387-98403-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Categories+for+the+Working+Mathematician&rft.place=Berlin%2C+New+York&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-0-387-98403-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation cs2"><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner, Charles W.</a>; <a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne, Kip</a>; <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</a> (1973), <a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Gravitation</a>, W. H. Freeman, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0344-0" title="Special:BookSources/978-0-7167-0344-0"><bdi>978-0-7167-0344-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pub=W.+H.+Freeman&rft.date=1973&rft.isbn=978-0-7167-0344-0&rft.aulast=Misner&rft.aufirst=Charles+W.&rft.au=Thorne%2C+Kip&rft.au=Wheeler%2C+John+Archibald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNaber2003" class="citation cs2">Naber, Gregory L. (2003), The geometry of Minkowski spacetime, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43235-9" title="Special:BookSources/978-0-486-43235-9"><bdi>978-0-486-43235-9</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=2044239">2044239</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+geometry+of+Minkowski+spacetime&rft.place=New+York&rft.pub=Dover+Publications&rft.date=2003&rft.isbn=978-0-486-43235-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2044239%23id-name%3DMR&rft.aulast=Naber&rft.aufirst=Gregory+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchönhageStrassen1971" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Arnold_Sch%C3%B6nhage" title="Arnold Schönhage">Schönhage, A.</a>; <a href="/wiki/Volker_Strassen" title="Volker Strassen">Strassen, Volker</a> (1971), "Schnelle Multiplikation großer Zahlen (Fast multiplication of big numbers)", Computing (in German), 7 (3–4): 281–292, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02242355">10.1007/bf02242355</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0010-485X">0010-485X</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9738629">9738629</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computing&rft.atitle=Schnelle+Multiplikation+gro%C3%9Fer+Zahlen+%28Fast+multiplication+of+big+numbers%29&rft.volume=7&rft.issue=%3Cspan+class%3D%22nowrap%22%3E3%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E281-%3C%2Fspan%3E292&rft.date=1971&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9738629%23id-name%3DS2CID&rft.issn=0010-485X&rft_id=info%3Adoi%2F10.1007%2Fbf02242355&rft.aulast=Sch%C3%B6nhage&rft.aufirst=A.&rft.au=Strassen%2C+Volker&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak1999" class="citation cs2"><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (1999), A Comprehensive Introduction to Differential Geometry (Volume Two), Houston, TX: Publish or Perish</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Comprehensive+Introduction+to+Differential+Geometry+%28Volume+Two%29&rft.place=Houston%2C+TX&rft.pub=Publish+or+Perish&rft.date=1999&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart1975" class="citation cs2"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (1975), <a rel="nofollow" class="external text" href="https://archive.org/details/galoistheory0000stew">Galois Theory</a>, <a href="/wiki/Chapman_and_Hall" class="mw-redirect" title="Chapman and Hall">Chapman and Hall</a> Mathematics Series, London: <a href="/wiki/Chapman_and_Hall" class="mw-redirect" title="Chapman and Hall">Chapman and Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-10800-6" title="Special:BookSources/978-0-412-10800-6"><bdi>978-0-412-10800-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois+Theory&rft.place=London&rft.series=Chapman+and+Hall+Mathematics+Series&rft.pub=Chapman+and+Hall&rft.date=1975&rft.isbn=978-0-412-10800-6&rft.aulast=Stewart&rft.aufirst=Ian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgaloistheory0000stew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaradarajan1974" class="citation cs2">Varadarajan, V. S. (1974), Lie groups, Lie algebras, and their representations, <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-535732-3" title="Special:BookSources/978-0-13-535732-3"><bdi>978-0-13-535732-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+groups%2C+Lie+algebras%2C+and+their+representations&rft.pub=Prentice+Hall&rft.date=1974&rft.isbn=978-0-13-535732-3&rft.aulast=Varadarajan&rft.aufirst=V.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWallace1992" class="citation cs2">Wallace, G.K. (Feb 1992), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070113155847/http://www.csc.ncsu.edu/faculty/rhee/export/papers/TheJPEGStillPictureCompressionStandard.pdf">"The JPEG still picture compression standard"</a> (PDF), IEEE Transactions on Consumer Electronics, 38 (1): xviii–xxxiv, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.318.4292">10.1.1.318.4292</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F30.125072">10.1109/30.125072</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0098-3063">0098-3063</a>, archived from <a rel="nofollow" class="external text" href="http://www.csc.ncsu.edu/faculty/rhee/export/papers/TheJPEGStillPictureCompressionStandard.pdf">the original</a> (PDF) on 2007-01-13, retrieved 2017-10-25</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Consumer+Electronics&rft.atitle=The+JPEG+still+picture+compression+standard&rft.volume=38&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3Exviii-%3C%2Fspan%3Exxxiv&rft.date=1992-02&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.318.4292%23id-name%3DCiteSeerX&rft.issn=0098-3063&rft_id=info%3Adoi%2F10.1109%2F30.125072&rft.aulast=Wallace&rft.aufirst=G.K.&rft_id=http%3A%2F%2Fwww.csc.ncsu.edu%2Ffaculty%2Frhee%2Fexport%2Fpapers%2FTheJPEGStillPictureCompressionStandard.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeibel1994" class="citation book cs1"><a href="/wiki/Charles_Weibel" title="Charles Weibel">Weibel, Charles A.</a> (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55987-4" title="Special:BookSources/978-0-521-55987-4"><bdi>978-0-521-55987-4</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=1269324">1269324</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/36131259">36131259</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+homological+algebra&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=info%3Aoclcnum%2F36131259&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1269324%23id-name%3DMR&rft.isbn=978-0-521-55987-4&rft.aulast=Weibel&rft.aufirst=Charles+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Vector_space&action=edit&section=35" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox 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title="wikibooks:Linear Algebra">Linear Algebra</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Vector_Spaces" class="extiw" title="wikibooks:Linear Algebra/Definition and Examples of Vector Spaces">Real vector spaces</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wikibooks-logo-en-noslogan.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></a></div> <div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Linear_Algebra" class="extiw" title="wikibooks:Linear Algebra">Linear Algebra</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Linear_Algebra/Vector_spaces" class="extiw" title="wikibooks:Linear Algebra/Vector spaces">Vector spaces</a></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Vector_space">"Vector space"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Vector+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DVector_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVector+space" class="Z3988"></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"> 78 languages </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"> Add topic </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-b766959bd-7jhvm","wgBackendResponseTime":135,"wgPageParseReport":{"limitreport":{"cputime":"1.832","walltime":"2.113","ppvisitednodes":{"value":22575,"limit":1000000},"postexpandincludesize":{"value":252544,"limit":2097152},"templateargumentsize":{"value":24556,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":32,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":313455,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1501.742 1 -total"," 26.86% 403.409 78 Template:Citation"," 17.92% 269.130 92 Template:Sfn"," 11.25% 168.939 197 Template:Math"," 7.00% 105.061 1 Template:Algebraic_structures"," 6.88% 103.375 1 Template:Sidebar_with_collapsible_lists"," 5.85% 87.883 301 Template:Main_other"," 5.70% 85.602 1 Template:Short_description"," 4.28% 64.242 2 Template:Pagetype"," 4.01% 60.277 1 Template:Authority_control"]},"scribunto":{"limitreport-timeusage":{"value":"0.958","limit":"10.000"},"limitreport-memusage":{"value":8519093,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREF\"] = 5,\n [\"CITEREFAntonRorres2010\"] = 1,\n [\"CITEREFArtin1991\"] = 1,\n [\"CITEREFAshcroftMermin1976\"] = 1,\n [\"CITEREFAtiyah1989\"] = 1,\n [\"CITEREFAtiyahMacdonald1969\"] = 1,\n [\"CITEREFBSE-3\"] = 1,\n [\"CITEREFBanach1922\"] = 1,\n [\"CITEREFBellavitis1833\"] = 1,\n [\"CITEREFBlass1984\"] = 1,\n [\"CITEREFBolzano1804\"] = 1,\n [\"CITEREFBourbaki1969\"] = 1,\n [\"CITEREFBourbaki1987\"] = 1,\n [\"CITEREFBourbaki1989\"] = 1,\n [\"CITEREFBourbaki1998\"] = 1,\n [\"CITEREFBourbaki2004\"] = 1,\n [\"CITEREFBraun1993\"] = 1,\n [\"CITEREFBrown1991\"] = 1,\n [\"CITEREFChoquet1966\"] = 1,\n [\"CITEREFCoxeter1987\"] = 1,\n [\"CITEREFDenneryKrzywicki1996\"] = 1,\n [\"CITEREFDorier1995\"] = 1,\n [\"CITEREFDudley1989\"] = 1,\n [\"CITEREFDunham2005\"] = 1,\n [\"CITEREFEisenbergGuy1979\"] = 1,\n [\"CITEREFEisenbud1995\"] = 1,\n [\"CITEREFEvans1998\"] = 1,\n [\"CITEREFFolland1992\"] = 1,\n [\"CITEREFFourier1822\"] = 1,\n [\"CITEREFGasquetWitomski1999\"] = 1,\n [\"CITEREFGoldrei1996\"] = 1,\n [\"CITEREFGrassmann1844\"] = 1,\n [\"CITEREFGrassmann2000\"] = 1,\n [\"CITEREFGriffiths1995\"] = 1,\n [\"CITEREFGrillet2007\"] = 1,\n [\"CITEREFGuo2021\"] = 1,\n [\"CITEREFHalmos1948\"] = 1,\n [\"CITEREFHalmos1974\"] = 1,\n [\"CITEREFHalpern1966\"] = 1,\n [\"CITEREFHamilton1853\"] = 1,\n [\"CITEREFHeil2011\"] = 1,\n [\"CITEREFHughes-HallettMcCallumGleason2013\"] = 1,\n [\"CITEREFHusemoller1994\"] = 1,\n [\"CITEREFIfeachorJervis2001\"] = 1,\n [\"CITEREFJain2001\"] = 1,\n [\"CITEREFJoshi1989\"] = 1,\n [\"CITEREFJost2005\"] = 1,\n [\"CITEREFKrantz1999\"] = 1,\n [\"CITEREFKreyszig1988\"] = 1,\n [\"CITEREFKreyszig1989\"] = 1,\n [\"CITEREFKreyszig1991\"] = 1,\n [\"CITEREFKreyszig1999\"] = 1,\n [\"CITEREFKreyszig2020\"] = 1,\n [\"CITEREFLang1983\"] = 1,\n [\"CITEREFLang1987\"] = 1,\n [\"CITEREFLang1993\"] = 1,\n [\"CITEREFLoomis2011\"] = 1,\n [\"CITEREFLuenberger1997\"] = 1,\n [\"CITEREFMac_Lane1998\"] = 1,\n [\"CITEREFMac_Lane1999\"] = 1,\n [\"CITEREFMeyer2000\"] = 1,\n [\"CITEREFMisnerThorneWheeler1973\"] = 1,\n [\"CITEREFMoore1995\"] = 1,\n [\"CITEREFMöbius1827\"] = 1,\n [\"CITEREFNaber2003\"] = 1,\n [\"CITEREFNicholson2018\"] = 1,\n [\"CITEREFPeano1888\"] = 1,\n [\"CITEREFRoman2005\"] = 1,\n [\"CITEREFRudin1991\"] = 1,\n [\"CITEREFSchönhageStrassen1971\"] = 1,\n [\"CITEREFSpindler1993\"] = 1,\n [\"CITEREFSpivak1999\"] = 1,\n [\"CITEREFSpringer2000\"] = 1,\n [\"CITEREFStewart1975\"] = 1,\n [\"CITEREFStollWong1968\"] = 1,\n [\"CITEREFTreves1967\"] = 1,\n [\"CITEREFVaradarajan1974\"] = 1,\n [\"CITEREFWallace1992\"] = 1,\n [\"CITEREFvan_der_Waerden1993\"] = 1,\n [\"vector_hyperplane\"] = 1,\n [\"vector_line\"] = 1,\n [\"vector_plane\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Algebraic structures\"] = 1,\n [\"Anchor\"] = 1,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 77,\n [\"Clear\"] = 1,\n [\"DEFAULTSORT:Vector Space\"] = 1,\n [\"Distinguish\"] = 1,\n [\"Em\"] = 1,\n [\"Good article\"] = 1,\n [\"Harvnb\"] = 2,\n [\"Harvtxt\"] = 7,\n [\"Lang Algebra\"] = 1,\n [\"Linear algebra\"] = 1,\n [\"Main\"] = 17,\n [\"Math\"] = 197,\n [\"Multiple image\"] = 1,\n [\"Mvar\"] = 64,\n [\"Narici Beckenstein Topological Vector Spaces\"] = 1,\n [\"Nowrap\"] = 3,\n [\"Redirect\"] = 1,\n [\"Refbegin\"] = 4,\n [\"Refend\"] = 4,\n [\"Reflist\"] = 2,\n [\"Schaefer Wolff Topological Vector Spaces\"] = 1,\n [\"Sfn\"] = 92,\n [\"Sfnm\"] = 4,\n [\"Short description\"] = 1,\n [\"Slink\"] = 2,\n [\"Springer\"] = 2,\n [\"Weibel IHA\"] = 1,\n [\"Wikibooks\"] = 2,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-b766959bd-829g7","timestamp":"20250214040707","ttl":2592000,"transientcontent":false}}});});</script> <script 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