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href="#Estruturas_sobre_o_espazo_euclidiano"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Estruturas sobre o espazo euclidiano</span> </div> </a> <button aria-controls="toc-Estruturas_sobre_o_espazo_euclidiano-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Mostrar ou agochar a subsección "Estruturas sobre o espazo euclidiano"</span> </button> <ul id="toc-Estruturas_sobre_o_espazo_euclidiano-sublist" class="vector-toc-list"> <li id="toc-O_espazo_euclidiano_como_espazo_métrico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#O_espazo_euclidiano_como_espazo_métrico"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>O espazo euclidiano como espazo métrico</span> </div> </a> <ul id="toc-O_espazo_euclidiano_como_espazo_métrico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-O_espazo_euclidiano_como_espazo_topolóxico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#O_espazo_euclidiano_como_espazo_topolóxico"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>O espazo euclidiano como espazo topolóxico</span> </div> </a> <ul id="toc-O_espazo_euclidiano_como_espazo_topolóxico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-O_espazo_euclidiano_como_espazo_vectorial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#O_espazo_euclidiano_como_espazo_vectorial"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>O espazo euclidiano como espazo vectorial</span> </div> </a> <ul id="toc-O_espazo_euclidiano_como_espazo_vectorial-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Espazo_euclidiano_de_dimensión_infinita" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Espazo_euclidiano_de_dimensión_infinita"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Espazo euclidiano de dimensión infinita</span> </div> </a> <ul id="toc-Espazo_euclidiano_de_dimensión_infinita-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notas</span> </div> </a> <ul id="toc-Notas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véxase_tamén" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véxase_tamén"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Véxase tamén</span> </div> </a> <button aria-controls="toc-Véxase_tamén-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Mostrar ou agochar a subsección "Véxase tamén"</span> </button> <ul id="toc-Véxase_tamén-sublist" class="vector-toc-list"> <li id="toc-Outros_artigos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outros_artigos"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Outros artigos</span> </div> </a> <ul id="toc-Outros_artigos-sublist" class="vector-toc-list"> </ul> </li> </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="Contidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Mostrar ou agochar a táboa de contidos" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostrar ou agochar a táboa de contidos</span> </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"><span class="mw-page-title-main">Espazo euclidiano</span></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="Ir a un artigo noutra lingua. Dispoñible en 64 linguas" > <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-64" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">64 linguas</span> </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/Euklidiese_ruimte" title="Euklidiese ruimte – afrikaans" lang="af" hreflang="af" data-title="Euklidiese ruimte" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Euklidischer_Raum" title="Euklidischer Raum – alemán suízo" lang="gsw" hreflang="gsw" data-title="Euklidischer Raum" data-language-autonym="Alemannisch" data-language-local-name="alemán suízo" class="interlanguage-link-target"><span>Alemannisch</span></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_%D8%A5%D9%82%D9%84%D9%8A%D8%AF%D9%8A" title="فضاء إقليدي – árabe" lang="ar" hreflang="ar" data-title="فضاء إقليدي" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_euclideu" title="Espaciu euclideu – asturiano" lang="ast" hreflang="ast" data-title="Espaciu euclideu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%93%D1%8B" title="Евклид арауығы – baxkir" lang="ba" hreflang="ba" data-title="Евклид арауығы" data-language-autonym="Башҡортса" data-language-local-name="baxkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%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="Евклидово пространство – búlgaro" lang="bg" hreflang="bg" data-title="Евклидово пространство" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%87%E0%A6%89%E0%A6%95%E0%A7%8D%E0%A6%B2%E0%A6%BF%E0%A6%A1%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8" title="ইউক্লিডীয় স্থান – bengalí" lang="bn" hreflang="bn" data-title="ইউক্লিডীয় স্থান" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_euclidi%C3%A0" title="Espai euclidià – catalán" lang="ca" hreflang="ca" data-title="Espai euclidià" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></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%DB%8C%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="بۆشاییی ئیقلیدسی – kurdo central" lang="ckb" hreflang="ckb" data-title="بۆشاییی ئیقلیدسی" data-language-autonym="کوردی" data-language-local-name="kurdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Eukleidovsk%C3%BD_prostor" title="Eukleidovský prostor – checo" lang="cs" hreflang="cs" data-title="Eukleidovský prostor" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D1%83%C3%A7%D0%BB%C4%83%D1%85%C4%95" title="Евклид уçлăхĕ – chuvaxo" lang="cv" hreflang="cv" data-title="Евклид уçлăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="chuvaxo" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_Euclidaidd" title="Gofod Euclidaidd – galés" lang="cy" hreflang="cy" data-title="Gofod Euclidaidd" data-language-autonym="Cymraeg" data-language-local-name="galés" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Euklidisk_rum" title="Euklidisk rum – dinamarqués" lang="da" hreflang="da" data-title="Euklidisk rum" data-language-autonym="Dansk" data-language-local-name="dinamarqués" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Euklidischer_Raum" title="Euklidischer Raum – alemán" lang="de" hreflang="de" data-title="Euklidischer Raum" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%BF%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Ευκλείδειος χώρος – grego" lang="el" hreflang="el" data-title="Ευκλείδειος χώρος" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euclidean_space" title="Euclidean space – inglés" lang="en" hreflang="en" data-title="Euclidean space" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/E%C5%ADklida_spaco" title="Eŭklida spaco – esperanto" lang="eo" hreflang="eo" data-title="Eŭklida spaco" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_eucl%C3%ADdeo" title="Espacio euclídeo – español" lang="es" hreflang="es" data-title="Espacio euclídeo" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Eukleidiline_ruum" title="Eukleidiline ruum – estoniano" lang="et" hreflang="et" data-title="Eukleidiline ruum" data-language-autonym="Eesti" data-language-local-name="estoniano" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Euklidear_espazio" title="Euklidear espazio – éuscaro" lang="eu" hreflang="eu" data-title="Euklidear espazio" data-language-autonym="Euskara" data-language-local-name="éuscaro" class="interlanguage-link-target"><span>Euskara</span></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%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="فضای اقلیدسی – persa" lang="fa" hreflang="fa" data-title="فضای اقلیدسی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Euklidinen_avaruus" title="Euklidinen avaruus – finés" lang="fi" hreflang="fi" data-title="Euklidinen avaruus" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_euclidien" title="Espace euclidien – francés" lang="fr" hreflang="fr" data-title="Espace euclidien" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></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%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99" title="מרחב אוקלידי – hebreo" lang="he" hreflang="he" data-title="מרחב אוקלידי" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AF%E0%A5%82%E0%A4%95%E0%A5%8D%E0%A4%B2%E0%A4%BF%E0%A4%A1%E0%A5%80%E0%A4%A8_%E0%A4%B8%E0%A4%AE%E0%A4%B7%E0%A5%8D%E0%A4%9F%E0%A4%BF" title="यूक्लिडीन समष्टि – hindi" lang="hi" hreflang="hi" data-title="यूक्लिडीन समष्टि" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Euklidski_prostor" title="Euklidski prostor – croata" lang="hr" hreflang="hr" data-title="Euklidski prostor" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euklideszi_t%C3%A9r_(line%C3%A1ris_algebra)" title="Euklideszi tér (lineáris algebra) – húngaro" lang="hu" hreflang="hu" data-title="Euklideszi tér (lineáris algebra)" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_Euklides" title="Ruang Euklides – indonesio" lang="id" hreflang="id" data-title="Ruang Euklides" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Euklidana_spaco" title="Euklidana spaco – ido" lang="io" hreflang="io" data-title="Euklidana spaco" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_euclideo" title="Spazio euclideo – italiano" lang="it" hreflang="it" data-title="Spazio euclideo" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A6%E3%83%BC%E3%82%AF%E3%83%AA%E3%83%83%E3%83%89%E7%A9%BA%E9%96%93" title="ユークリッド空間 – xaponés" lang="ja" hreflang="ja" data-title="ユークリッド空間" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D0%BA%D0%B5%D2%A3%D1%96%D1%81%D1%82%D1%96%D0%B3%D1%96" title="Евклид кеңістігі – kazako" lang="kk" hreflang="kk" data-title="Евклид кеңістігі" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간 – coreano" lang="ko" hreflang="ko" data-title="유클리드 공간" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Euklidin%C4%97_erdv%C4%97" title="Euklidinė erdvė – lituano" lang="lt" hreflang="lt" data-title="Euklidinė erdvė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Евклидов простор – macedonio" lang="mk" hreflang="mk" data-title="Евклидов простор" data-language-autonym="Македонски" data-language-local-name="macedonio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AF%E0%B5%82%E0%B4%95%E0%B5%8D%E0%B4%B2%E0%B4%BF%E0%B4%A1%E0%B4%BF%E0%B4%AF%E0%B5%BB_%E0%B4%B8%E0%B5%8D%E0%B4%AA%E0%B5%86%E0%B4%AF%E0%B5%8D%E0%B4%B8%E0%B5%8D" title="യൂക്ലിഡിയൻ സ്പെയ്സ് – malabar" lang="ml" hreflang="ml" data-title="യൂക്ലിഡിയൻ സ്പെയ്സ്" data-language-autonym="മലയാളം" data-language-local-name="malabar" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%B8%D0%B9%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD_%D0%B7%D0%B0%D0%B9" title="Евклидийн орон зай – mongol" lang="mn" hreflang="mn" data-title="Евклидийн орон зай" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_Euclides" title="Ruang Euclides – malaio" lang="ms" hreflang="ms" data-title="Ruang Euclides" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9A%E1%80%B0%E1%80%80%E1%80%9C%E1%80%85%E1%80%BA%E1%80%92%E1%80%BA_%E1%80%85%E1%80%95%E1%80%B1%E1%80%B7%E1%80%85%E1%80%BA" title="ယူကလစ်ဒ် စပေ့စ် – birmano" lang="my" hreflang="my" data-title="ယူကလစ်ဒ် စပေ့စ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Euclidische_ruimte" title="Euclidische ruimte – neerlandés" lang="nl" hreflang="nl" data-title="Euclidische ruimte" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Euklidsk_rom" title="Euklidsk rom – noruegués bokmål" lang="nb" hreflang="nb" data-title="Euklidsk rom" data-language-autonym="Norsk bokmål" data-language-local-name="noruegués bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AF%E0%A9%81%E0%A8%95%E0%A8%B2%E0%A8%BF%E0%A8%A1%E0%A9%80%E0%A8%85%E0%A8%A8_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਯੁਕਲਿਡੀਅਨ ਸਪੇਸ – panxabí" lang="pa" hreflang="pa" data-title="ਯੁਕਲਿਡੀਅਨ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panxabí" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_euklidesowa" title="Przestrzeń euklidesowa – polaco" lang="pl" hreflang="pl" data-title="Przestrzeń euklidesowa" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C_%D8%B3%D9%BE%DB%8C%D8%B3" title="اقلیدسی سپیس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="اقلیدسی سپیس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_euclidiano" title="Espaço euclidiano – portugués" lang="pt" hreflang="pt" data-title="Espaço euclidiano" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_euclidian" title="Spațiu euclidian – romanés" lang="ro" hreflang="ro" data-title="Spațiu euclidian" data-language-autonym="Română" data-language-local-name="romanés" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2%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="Евклидово пространство – ruso" lang="ru" hreflang="ru" data-title="Евклидово пространство" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Euklidski_prostor" title="Euklidski prostor – serbocroata" lang="sh" hreflang="sh" data-title="Euklidski prostor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BA%E0%B7%94%E0%B6%9A%E0%B7%8A%E0%B6%BD%E0%B7%92%E0%B6%A9%E0%B7%92%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%94_%E0%B6%85%E0%B7%80%E0%B6%9A%E0%B7%8F%E0%B7%81%E0%B6%BA" title="යුක්ලිඩියානු අවකාශය – cingalés" lang="si" hreflang="si" data-title="යුක්ලිඩියානු අවකාශය" data-language-autonym="සිංහල" data-language-local-name="cingalés" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Euclidean_space" title="Euclidean space – Simple English" lang="en-simple" hreflang="en-simple" data-title="Euclidean space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Evklidski_prostor" title="Evklidski prostor – esloveno" lang="sl" hreflang="sl" data-title="Evklidski prostor" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hap%C3%ABsira_Euklidiane" title="Hapësira Euklidiane – albanés" lang="sq" hreflang="sq" data-title="Hapësira Euklidiane" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D1%83%D0%BA%D0%BB%D0%B8%D0%B4%D0%BE%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Еуклидов простор – serbio" lang="sr" hreflang="sr" data-title="Еуклидов простор" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Euklidiskt_rum" title="Euklidiskt rum – sueco" lang="sv" hreflang="sv" data-title="Euklidiskt rum" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AF%E0%AF%82%E0%AE%95%E0%AF%8D%E0%AE%B3%E0%AE%BF%E0%AE%9F%E0%AE%BF%E0%AE%AF_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF" title="யூக்ளிடிய வெளி – támil" lang="ta" hreflang="ta" data-title="யூக்ளிடிய வெளி" data-language-autonym="தமிழ்" data-language-local-name="támil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espasyong_Euclides" title="Espasyong Euclides – tagalo" lang="tl" hreflang="tl" data-title="Espasyong Euclides" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96klid_uzay%C4%B1" title="Öklid uzayı – turco" lang="tr" hreflang="tr" data-title="Öklid uzayı" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D0%B8%D0%B4_%D1%84%D3%99%D0%B7%D0%B0%D1%81%D1%8B" title="Евклид фәзасы – tártaro" lang="tt" hreflang="tt" data-title="Евклид фәзасы" data-language-autonym="Татарча / tatarça" data-language-local-name="tártaro" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%B2%D0%BA%D0%BB%D1%96%D0%B4%D1%96%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Евклідів простір – ucraíno" lang="uk" hreflang="uk" data-title="Евклідів простір" data-language-autonym="Українська" data-language-local-name="ucraíno" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yevklid_fazosi" title="Yevklid fazosi – uzbeko" lang="uz" hreflang="uz" data-title="Yevklid fazosi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeko" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_Euclid" title="Không gian Euclid – vietnamita" lang="vi" hreflang="vi" data-title="Không gian Euclid" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间 – chinés wu" lang="wuu" hreflang="wuu" data-title="欧几里得空间" data-language-autonym="吴语" data-language-local-name="chinés wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间 – chinés" lang="zh" hreflang="zh" data-title="欧几里得空间" data-language-autonym="中文" data-language-local-name="chinés" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AD%90%E5%B9%BE%E9%87%8C%E5%BE%97%E7%A9%BA%E9%96%93" title="歐幾里得空間 – cantonés" lang="yue" hreflang="yue" data-title="歐幾里得空間" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">agochar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Na Galipedia, a Wikipedia en galego.</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="gl" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Coord_system_CA_0.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/250px-Coord_system_CA_0.svg.png" decoding="async" width="250" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/375px-Coord_system_CA_0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Coord_system_CA_0.svg/500px-Coord_system_CA_0.svg.png 2x" data-file-width="620" data-file-height="600" /></a><figcaption>Representación gráfica de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>.</figcaption></figure> <p>O <b>espazo euclidiano </b>é un espazo xeométrico no que se satisfán os axiomas de Euclides da xeometría. A recta real, o <a href="/wiki/Plano_(xeometr%C3%ADa)" title="Plano (xeometría)">plano euclidiano</a> e o espazo <a href="/wiki/Espazo_tridimensional" title="Espazo tridimensional">tridimensional</a> da <a href="/wiki/Xeometr%C3%ADa_euclidiana" title="Xeometría euclidiana">xeometría euclidiana</a> son casos especiais de espazos euclidiano de dimensións 1, 2 e 3 respectivamente. O concepto abstracto de espazo euclídeo xeneraliza esas construcións a máis dimensións. Un espazo euclidiano é un espazo vectorial completo dotado dun produto interno (o cal o converte ademais nun <a href="/w/index.php?title=Espazo_normado&action=edit&redlink=1" class="new" title="Espazo normado (a páxina aínda non existe)">espazo normado</a>, un <a href="/wiki/Espazo_m%C3%A9trico" title="Espazo métrico">espazo métrico</a> e unha <a href="/w/index.php?title=Variedade_riemanniana&action=edit&redlink=1" class="new" title="Variedade riemanniana (a páxina aínda non existe)">variedade riemanniana</a> ao mesmo tempo). </p><p>O termo <b>euclidiano</b> emprégase para distinguir estes espazos dos espazos "curvos" das <a href="/wiki/Xeometr%C3%ADa_non_euclidiana" title="Xeometría non euclidiana">xeometrías non euclidianas</a> e do espazo da <a href="/wiki/Teor%C3%ADa_da_relatividade" title="Teoría da relatividade">teoría da relatividade</a> de <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>. Para destacar o fe<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0" /> </mrow> <annotation encoding="application/x-tex">{\displaystyle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0; height:0.343ex;" alt="{\displaystyle }"></span>ito d<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0" /> </mrow> <annotation encoding="application/x-tex">{\displaystyle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0; height:0.343ex;" alt="{\displaystyle }"></span>e que un espazo euclidiano pode posuír <i>n</i> dimensións, adóitase falar de "espazo euclidiano <i>n</i>-dimensional" (denotado <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {E} ^{n},E^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {E} ^{n},E^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a10d48cfe222973faeaf5ae56d580fdf387810a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.752ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \mathbb {E} ^{n},E^{n}}"></span>, ou mesmo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1a276509225c16a268cab878dbbd1856d919d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.151ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \mathbb {R} ^{n}}"></span>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introdución"><span id="Introduci.C3.B3n"></span>Introdución</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=1" title="Editar a sección: «Introdución»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=1" title="Editar o código fonte da sección: Introdución"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un espazo euclidiano de dimensión finita é un <a href="/w/index.php?title=Espazo_vectorial_normado&action=edit&redlink=1" class="new" title="Espazo vectorial normado (a páxina aínda non existe)">espazo vectorial normado</a> sobre os números reais de <a href="/wiki/Dimensi%C3%B3n" title="Dimensión">dimensión</a> finita, no que a norma é a asociada ao <a href="/wiki/Produto_escalar" title="Produto escalar">produto escalar</a> ordinario. Para cada <a href="/wiki/N%C3%BAmero_enteiro" title="Número enteiro">número enteiro</a> non negativo n, o espazo euclidiano n-dimensional represéntase polo símbolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> e é o conxunto de todas as <a href="/wiki/Tupla" title="Tupla">tuplas</a> ordenadas </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d8b3ec62633086002489cad8df4214e9585880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.337ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}"></span> </p> </blockquote> <p>onde cada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> é un número real, xunto coa <a href="/wiki/Distancia" title="Distancia">función distancia</a> entre dous puntos (<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>) e (<i>y</i><sub>1</sub>, ..., <i>y</i><sub><i>n</i></sub>) definida pola fórmula: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</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> <mo>=</mo> <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"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be1e92c82a80285817604354275b99fb2bb5d4da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.516ex; height:7.509ex;" alt="{\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}"></span> </p> </blockquote> <p>Esta función distancia é unha xeneralización do <a href="/wiki/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras">teorema de Pitágoras</a> e denomínase distancia euclidiana. O feito de que se definise unha distancia permite definir outros conceptos métricos como o de medida de Lebesgue (que permite á súa vez definir a <a href="/w/index.php?title=Lonxitude_dunha_curva&action=edit&redlink=1" class="new" title="Lonxitude dunha curva (a páxina aínda non existe)">lonxitude dunha curva</a> (1-<a href="/wiki/Volume_(magnitude)" title="Volume (magnitude)">volume</a>), as nocións de <a href="/wiki/%C3%81rea" title="Área">área</a> (2-volume), volume (3-volume) e cando o espazo ten dimensión superior a 3 n-volume (para <i>n</i> > 3). </p><p>Ademais poden definirse ángulos, ao poder falar de <a href="/wiki/Proxecci%C3%B3n" title="Proxección">proxectar</a> unha lonxitude recta sobre a dirección doutra lonxitude recta non paralela, polo que o ángulo entre dúas rectas <i>r</i><sub>1</sub> e <i>r</i><sub>2</sub> con vectores unitarios tanxentes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4239c3b98880c8455cfb0dbdecce01fdecc6c92b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \mathbf {n} _{1}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {n} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d7ec4ad2fddcf79be8979fd8d44c45924d69a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \mathbf {n} _{2}}"></span> se pode definir como: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arccos \left(\mathbf {n} _{1}\cdot \mathbf {n} _{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arccos \left(\mathbf {n} _{1}\cdot \mathbf {n} _{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fba7d1db96d16bf7fe5c6eb7745cda7efedc0c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.974ex; height:2.843ex;" alt="{\displaystyle \theta =\arccos \left(\mathbf {n} _{1}\cdot \mathbf {n} _{2}\right)}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Estruturas_sobre_o_espazo_euclidiano">Estruturas sobre o espazo euclidiano</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=2" title="Editar a sección: «Estruturas sobre o espazo euclidiano»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=2" title="Editar o código fonte da sección: Estruturas sobre o espazo euclidiano"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os espazos euclidianos e as súas propiedades serviron de base para xerar gran cantidade de conceptos matemáticos relacionados coa <a href="/wiki/Xeometr%C3%ADa_anal%C3%ADtica" title="Xeometría analítica">xeometría analítica</a>, a topoloxía, a álxebra e o cálculo. Aínda que o espazo euclidiano adoita ser introducido, por razóns didácticas, como <a href="/wiki/Espazo_vectorial" title="Espazo vectorial">espazo vectorial</a>, en realidade sobre el pódense definir moitas máis estruturas. O espazo euclídeo é ademais dun espazo vectorial un caso de: </p> <ul><li>Un <a href="/wiki/Espazo_de_Hilbert" title="Espazo de Hilbert">espazo de Hilbert</a> de dimensión finita, co produto escalar ordinario.</li> <li>Un <a href="/w/index.php?title=Espazo_de_Banach&action=edit&redlink=1" class="new" title="Espazo de Banach (a páxina aínda non existe)">espazo de Banach</a> de dimensión finita, coa norma inducida polo produto escalar interior.</li> <li>Un <a href="/wiki/Espazo_m%C3%A9trico" title="Espazo métrico">espazo métrico</a> completo, coa distancia inducida pola norma anterior.</li> <li>Un <a href="/wiki/Espazo_topol%C3%B3xico" title="Espazo topolóxico">espazo topolóxico</a>, inducido pola métrica euclídea.</li> <li>Un grupo de Lie, coa operación de adición.</li> <li>Unha <a href="/wiki/%C3%81lxebra_de_Lie" title="Álxebra de Lie">álxebra de Lie</a> co produto vectorial.</li></ul> <div class="mw-heading mw-heading3"><h3 id="O_espazo_euclidiano_como_espazo_métrico"><span id="O_espazo_euclidiano_como_espazo_m.C3.A9trico"></span>O espazo euclidiano como espazo métrico</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=3" title="Editar a sección: «O espazo euclidiano como espazo métrico»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=3" title="Editar o código fonte da sección: O espazo euclidiano como espazo métrico"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Por definición, <i>E</i><sup> <i>n</i></sup> é un espazo métrico, e é polo tanto tamén un <a href="/wiki/Topolox%C3%ADa" title="Topoloxía">espazo topolóxico</a>; é o exemplo prototípico dunha n-variedade, e é de feito unha <i>n</i>-variedade diferenciable. Para <i>n</i> ≠ 4, calqu<i>e</i>ra <i>n</i>-variedade diferenciable que sexa homeomorfa a <i>E</i><sup> <i>n</i></sup> é tamén difeomorfa a ela. O feito sorprendente é que isto non é certo tamén para <i>n</i> = 4, o que foi probado por Simon Donaldson no ano 1982; os contraexemplos chámanse <a href="/w/index.php?title=4-variedade&action=edit&redlink=1" class="new" title="4-variedade (a páxina aínda non existe)">4-espazos</a> exóticos (ou falsos). </p> <div class="mw-heading mw-heading3"><h3 id="O_espazo_euclidiano_como_espazo_topolóxico"><span id="O_espazo_euclidiano_como_espazo_topol.C3.B3xico"></span>O espazo euclidiano como espazo topolóxico</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=4" title="Editar a sección: «O espazo euclidiano como espazo topolóxico»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=4" title="Editar o código fonte da sección: O espazo euclidiano como espazo topolóxico"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pódese dicir moito sobre a <a href="/wiki/Topolox%C3%ADa" title="Topoloxía">topoloxía</a> de <i>E</i><sup> <i>n</i></sup>. Un resultado importante, a invariancia do dominio de Brouwer, é o de que calquera subconxunto de <i>E</i><sup> <i>n</i></sup> que sexa homeomorfo a un subconxunto aberto de <i>E</i><sup> <i>n</i></sup> é en si mesmo aberto. Como consecuencia inmediata disto tense que E m non é homeomorfo a <i>E</i><sup> <i>n</i></sup> se <i>m</i> ≠ <i>n</i>, un resultado intuitivamente "obvio" que con todo non é doado de demostrar. </p> <div class="mw-heading mw-heading3"><h3 id="O_espazo_euclidiano_como_espazo_vectorial">O espazo euclidiano como espazo vectorial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=5" title="Editar a sección: «O espazo euclidiano como espazo vectorial»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=5" title="Editar o código fonte da sección: O espazo euclidiano como espazo vectorial"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O n-espazo euclidiano pódese considerar tamén como un <a href="/wiki/Espazo_vectorial" title="Espazo vectorial">espazo vectorial</a> n-dimensional real, de feito un <a href="/wiki/Espazo_de_Hilbert" title="Espazo de Hilbert">espazo de Hilbert</a>, de maneira natural. O <a href="/wiki/Produto_escalar" title="Produto escalar">produto escalar</a>, de <b>x</b> = (<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub>) e y = (<i>y</i><sub>1</sub>,...,<i>y</i><sub><i>n</i></sub>) está dado por: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}}"> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \cdot \mathbf {y} =\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a71ba4838c7a2c83743b73f8deea6c5bd76b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.813ex; height:6.843ex;" alt="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Espazo_euclidiano_de_dimensión_infinita"><span id="Espazo_euclidiano_de_dimensi.C3.B3n_infinita"></span>Espazo euclidiano de dimensión infinita</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=6" title="Editar a sección: «Espazo euclidiano de dimensión infinita»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=6" title="Editar o código fonte da sección: Espazo euclidiano de dimensión infinita"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os espazos euclidianos considerados usualmente ten unha <a href="/w/index.php?title=Dimensi%C3%B3n_topol%C3%B3xica&action=edit&redlink=1" class="new" title="Dimensión topolóxica (a páxina aínda non existe)">dimensión topolóxica</a> finita. Iso fai que sexan localmente <a href="/wiki/Espazo_compacto" title="Espazo compacto">compactos</a>. Con todo, é posíbel concibir estruturas de dimensión infinita que teñan propiedades análogas aos espazos euclidianos, polo que a extensión á dimensión infinita da noción de espazo euclidiano é posíbel cunhas poucas precaucións.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> En primeiro lugar pódese considerar o conxunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0881ff53d61be4b3ed806e41af0d2ffba29c09b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.933ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\omega }}"></span> definido como: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\omega }=\{(x_{1},x_{2},\dots )|\forall i:x_{i}\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</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 stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\omega }=\{(x_{1},x_{2},\dots )|\forall i:x_{i}\in \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54dbeb1b9b8f2fdcc33260190bf93b58d3328abb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.051ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} ^{\omega }=\{(x_{1},x_{2},\dots )|\forall i:x_{i}\in \mathbb {R} \}}"></span> </p> </blockquote> <p>É dicir este conxunto é o <a href="/wiki/Produto_cartesiano" title="Produto cartesiano">produto cartesiano</a> dun número <a href="/wiki/Conxunto_cont%C3%A1bel" class="mw-redirect" title="Conxunto contábel">infinito numerable</a> de copias de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. Con todo o conxunto de todas esas tuplas infinitas non ten a estrutura de espazo euclidiano porque non se pode dotar dunha <a href="/w/index.php?title=Norma_euclidiana&action=edit&redlink=1" class="new" title="Norma euclidiana (a páxina aínda non existe)">norma euclidiana</a> axeitada. Por exemplo as tuplas: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =(1,1,1,\dots ),{\text{ ó }}\mathbf {y} =(1,2,3,4,\dots )}"> <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>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> ó </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(1,1,1,\dots ),{\text{ ó }}\mathbf {y} =(1,2,3,4,\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c855df83a2a167030268d004f2cb4762d21a6c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.071ex; height:3.343ex;" alt="{\displaystyle \mathbf {x} =(1,1,1,\dots ),{\text{ ó }}\mathbf {y} =(1,2,3,4,\dots )}"></span> </p> </blockquote> <p>non representan vectores cunha suma de compoñentes ao cadrado que sexa un número real finito. Por esta razón considérase o subconxunto: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\infty }=\{(x_{1},x_{2},\dots )|\forall i:x_{i}\in \mathbb {R} ,\ \lim _{N\to \infty }\sum _{n=1}^{N}|x_{n}|^{2}<\infty \}\varsubsetneq \mathbb {R} ^{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</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 stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> <mo class="MJX-variant">⊊</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\infty }=\{(x_{1},x_{2},\dots )|\forall i:x_{i}\in \mathbb {R} ,\ \lim _{N\to \infty }\sum _{n=1}^{N}|x_{n}|^{2}<\infty \}\varsubsetneq \mathbb {R} ^{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d930a595949dda52cbedc06de4d4698adc5eee4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.935ex; height:7.343ex;" alt="{\displaystyle E_{\infty }=\{(x_{1},x_{2},\dots )|\forall i:x_{i}\in \mathbb {R} ,\ \lim _{N\to \infty }\sum _{n=1}^{N}|x_{n}|^{2}<\infty \}\varsubsetneq \mathbb {R} ^{\omega }}"></span> </p> </blockquote> <p>Este espazo vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68cacbe04c10fce753db60f346f92a34e1567d1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.591ex; height:2.509ex;" alt="{\displaystyle E_{\infty }}"></span> comparte a maior parte dos espazos euclidianos finitodimensionales e polo tanto pode considerarse un espazo euclidiano infinitodimensional; a principal propiedade é que o espazo euclidiano infinitodimensional a diferenza das súas versións finitodimensionais non é un espazo localmente compacto. </p> <div class="mw-heading mw-heading2"><h2 id="Notas">Notas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=7" title="Editar a sección: «Notas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=7" title="Editar o código fonte da sección: Notas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://link.springer.com/chapter/10.1007/978-1-4684-3926-7_6"><i>Infinite-Dimensional Euclidean Spaces.</i></a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Véxase_tamén"><span id="V.C3.A9xase_tam.C3.A9n"></span>Véxase tamén</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=8" title="Editar a sección: «Véxase tamén»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=8" title="Editar o código fonte da sección: Véxase tamén"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Outros_artigos">Outros artigos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Espazo_euclidiano&veaction=edit&section=9" title="Editar a sección: «Outros artigos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Espazo_euclidiano&action=edit&section=9" title="Editar o código fonte da sección: Outros artigos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Xeometr%C3%ADa_euclidiana" title="Xeometría euclidiana">Xeometría euclidiana</a></li> <li><a href="/wiki/Xeometr%C3%ADa_anal%C3%ADtica" title="Xeometría analítica">Xeometría analítica</a></li> <li><a href="/w/index.php?title=Medida_de_Lebesgue&action=edit&redlink=1" class="new" title="Medida de Lebesgue (a páxina aínda non existe)">Medida de Lebesgue</a></li></ul> <div role="navigation" class="navbox" aria-labelledby="Control_de_autoridades" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_de_autoridades" scope="row" class="navbox-group" style="width:1%;width: 12%; 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