Normált tér – Wikipédia

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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">Tartalomjegyzék</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">elrejtés</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">Bevezető</div> </a> </li> <li id="toc-Definíció" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definíció"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definíció</span> </div> </a> <ul id="toc-Definíció-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Példák" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Példák"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Példák</span> </div> </a> <ul id="toc-Példák-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tulajdonságok" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tulajdonságok"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Tulajdonságok</span> </div> </a> <button aria-controls="toc-Tulajdonságok-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>A(z) Tulajdonságok alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Tulajdonságok-sublist" class="vector-toc-list"> <li id="toc-Kapcsolat_a_metrikus_terekkel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kapcsolat_a_metrikus_terekkel"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Kapcsolat a metrikus terekkel</span> </div> </a> <ul id="toc-Kapcsolat_a_metrikus_terekkel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ekvivalens_normák" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ekvivalens_normák"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Ekvivalens normák</span> </div> </a> <ul id="toc-Ekvivalens_normák-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normált_terek_szorzata" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normált_terek_szorzata"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Normált terek szorzata</span> </div> </a> <ul id="toc-Normált_terek_szorzata-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Speciális_esetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Speciális_esetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Speciális esetek</span> </div> </a> <button aria-controls="toc-Speciális_esetek-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>A(z) Speciális esetek alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Speciális_esetek-sublist" class="vector-toc-list"> <li id="toc-Skalárszorzatos_vektorterek" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Skalárszorzatos_vektorterek"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Skalárszorzatos vektorterek</span> </div> </a> <ul id="toc-Skalárszorzatos_vektorterek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teljes_terek" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teljes_terek"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Teljes terek</span> </div> </a> <ul id="toc-Teljes_terek-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Általánosítások" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Általánosítások"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Általánosítások</span> </div> </a> <button aria-controls="toc-Általánosítások-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>A(z) Általánosítások alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Általánosítások-sublist" class="vector-toc-list"> <li id="toc-Félnormált_terek" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Félnormált_terek"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Félnormált terek</span> </div> </a> <ul id="toc-Félnormált_terek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metrikus_és_topologikus_terek" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metrikus_és_topologikus_terek"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Metrikus és topologikus terek</span> </div> </a> <ul id="toc-Metrikus_és_topologikus_terek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abszolútértékes_testek" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Abszolútértékes_testek"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Abszolútértékes testek</span> </div> </a> <ul id="toc-Abszolútértékes_testek-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Jegyzetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jegyzetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Források" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Források"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Források</span> </div> </a> <ul id="toc-Források-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fordítás" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fordítás"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Fordítás</span> </div> </a> <ul id="toc-Fordítás-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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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">Normált tér</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="Ugrás egy más nyelvű szócikkre. Elérhető 32 nyelven" > <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-32" 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">32 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Normed_vector_space" title="Normed vector space – angol" lang="en" hreflang="en" data-title="Normed vector space" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</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_%D9%85%D8%AA%D8%AC%D9%87%D9%8A_%D9%85%D8%B9%D9%8A%D8%A7%D8%B1%D9%8A" title="فضاء متجهي معياري – arab" lang="ar" hreflang="ar" data-title="فضاء متجهي معياري" data-language-autonym="العربية" data-language-local-name="arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_vectorial_normat" title="Espai vectorial normat – katalán" lang="ca" hreflang="ca" data-title="Espai vectorial normat" data-language-autonym="Català" data-language-local-name="katalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Normovan%C3%BD_line%C3%A1rn%C3%AD_prostor" title="Normovaný lineární prostor – cseh" lang="cs" hreflang="cs" data-title="Normovaný lineární prostor" data-language-autonym="Čeština" data-language-local-name="cseh" 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%9D%D0%BE%D1%80%D0%BC%C4%83%D0%BB%D0%B0%D0%B2%D0%BB%C4%83_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Нормăлавлă уçлăх – csuvas" lang="cv" hreflang="cv" data-title="Нормăлавлă уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="csuvas" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Normeret_vektorrum" title="Normeret vektorrum – dán" lang="da" hreflang="da" data-title="Normeret vektorrum" data-language-autonym="Dansk" data-language-local-name="dán" 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/Normierter_Raum" title="Normierter Raum – német" lang="de" hreflang="de" data-title="Normierter Raum" data-language-autonym="Deutsch" data-language-local-name="német" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_vectorial_normado" title="Espacio vectorial normado – spanyol" lang="es" hreflang="es" data-title="Espacio vectorial normado" data-language-autonym="Español" data-language-local-name="spanyol" 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/Normeeritud_ruum" title="Normeeritud ruum – észt" lang="et" hreflang="et" data-title="Normeeritud ruum" data-language-autonym="Eesti" data-language-local-name="észt" class="interlanguage-link-target"><span>Eesti</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%A8%D8%B1%D8%AF%D8%A7%D8%B1%DB%8C_%D9%86%D8%B1%D9%85%E2%80%8C%D8%AF%D8%A7%D8%B1" title="فضای برداری نرم‌دار – perzsa" lang="fa" hreflang="fa" data-title="فضای برداری نرم‌دار" data-language-autonym="فارسی" data-language-local-name="perzsa" 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/Normiavaruus" title="Normiavaruus – finn" lang="fi" hreflang="fi" data-title="Normiavaruus" data-language-autonym="Suomi" data-language-local-name="finn" 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_vectoriel_norm%C3%A9" title="Espace vectoriel normé – francia" lang="fr" hreflang="fr" data-title="Espace vectoriel normé" data-language-autonym="Français" data-language-local-name="francia" 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%A0%D7%95%D7%A8%D7%9E%D7%99" title="מרחב נורמי – héber" lang="he" hreflang="he" data-title="מרחב נורמי" data-language-autonym="עברית" data-language-local-name="héber" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_vektor_bernorma" title="Ruang vektor bernorma – indonéz" lang="id" hreflang="id" data-title="Ruang vektor bernorma" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéz" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sta%C3%B0la%C3%B0_vigurr%C3%BAm" title="Staðlað vigurrúm – izlandi" lang="is" hreflang="is" data-title="Staðlað vigurrúm" data-language-autonym="Íslenska" data-language-local-name="izlandi" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_normato" title="Spazio normato – olasz" lang="it" hreflang="it" data-title="Spazio normato" data-language-autonym="Italiano" data-language-local-name="olasz" 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%8E%E3%83%AB%E3%83%A0%E7%B7%9A%E5%9E%8B%E7%A9%BA%E9%96%93" title="ノルム線型空間 – japán" lang="ja" hreflang="ja" data-title="ノルム線型空間" data-language-autonym="日本語" data-language-local-name="japán" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%85%B8%EB%A6%84_%EA%B3%B5%EA%B0%84" title="노름 공간 – koreai" lang="ko" hreflang="ko" data-title="노름 공간" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Genormeerde_vectorruimte" title="Genormeerde vectorruimte – holland" lang="nl" hreflang="nl" data-title="Genormeerde vectorruimte" data-language-autonym="Nederlands" data-language-local-name="holland" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A8%E0%A9%8C%E0%A8%B0%E0%A8%AE%E0%A8%A1_%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="ਨੌਰਮਡ ਵੈਕਟਰ ਸਪੇਸ – pandzsábi" lang="pa" hreflang="pa" data-title="ਨੌਰਮਡ ਵੈਕਟਰ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandzsábi" 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_unormowana" title="Przestrzeń unormowana – lengyel" lang="pl" hreflang="pl" data-title="Przestrzeń unormowana" data-language-autonym="Polski" data-language-local-name="lengyel" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7os_normados" title="Espaços normados – portugál" lang="pt" hreflang="pt" data-title="Espaços normados" data-language-autonym="Português" data-language-local-name="portugál" 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_vectorial_normat" title="Spațiu vectorial normat – román" lang="ro" hreflang="ro" data-title="Spațiu vectorial normat" data-language-autonym="Română" data-language-local-name="román" 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%9D%D0%BE%D1%80%D0%BC%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%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="Нормированное пространство – orosz" lang="ru" hreflang="ru" data-title="Нормированное пространство" data-language-autonym="Русский" data-language-local-name="orosz" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Normovan%C3%BD_line%C3%A1rny_priestor" title="Normovaný lineárny priestor – szlovák" lang="sk" hreflang="sk" data-title="Normovaný lineárny priestor" data-language-autonym="Slovenčina" data-language-local-name="szlovák" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Normerat_rum" title="Normerat rum – svéd" lang="sv" hreflang="sv" data-title="Normerat rum" data-language-autonym="Svenska" data-language-local-name="svéd" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Normlu_vekt%C3%B6r_uzay%C4%B1" title="Normlu vektör uzayı – török" lang="tr" hreflang="tr" data-title="Normlu vektör uzayı" data-language-autonym="Türkçe" data-language-local-name="török" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Нормований простір – ukrán" lang="uk" hreflang="uk" data-title="Нормований простір" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Spassio_norm%C3%A0" title="Spassio normà – velencei" lang="vec" hreflang="vec" data-title="Spassio normà" data-language-autonym="Vèneto" data-language-local-name="velencei" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_%C4%91%E1%BB%8Bnh_chu%E1%BA%A9n" title="Không gian định chuẩn – vietnámi" lang="vi" hreflang="vi" data-title="Không gian định chuẩn" data-language-autonym="Tiếng Việt" data-language-local-name="vietnámi" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B3%A6%E7%AF%84%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" title="賦範向量空間 – kínai" lang="zh" hreflang="zh" data-title="賦範向量空間" data-language-autonym="中文" data-language-local-name="kínai" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%AF%84%E7%A9%BA%E9%96%93" title="範空間 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="範空間" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q726210#sitelinks-wikipedia" title="Nyelvközi hivatkozások szerkesztése" class="wbc-editpage">Hivatkozások szerkesztése</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Névterek"> <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/Norm%C3%A1lt_t%C3%A9r" title="A lap megtekintése [c]" accesskey="c"><span>Szócikk</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Vita:Norm%C3%A1lt_t%C3%A9r" rel="discussion" title="Az oldal tartalmának megvitatása [t]" accesskey="t"><span>Vitalap</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Nyelvvariáns váltása" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">magyar</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Nézetek"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Norm%C3%A1lt_t%C3%A9r"><span>Olvasás</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=history" title="A lap korábbi változatai [h]" accesskey="h"><span>Laptörténet</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eszközök" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eszközök</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eszközök</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">elrejtés</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="További lehetőségek" > <div class="vector-menu-heading"> Műveletek </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Norm%C3%A1lt_t%C3%A9r"><span>Olvasás</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=history"><span>Laptörténet</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Általános </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Mi_hivatkozik_erre/Norm%C3%A1lt_t%C3%A9r" title="Az erre a lapra hivatkozó más lapok listája [j]" accesskey="j"><span>Mi hivatkozik erre?</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Kapcsol%C3%B3d%C3%B3_v%C3%A1ltoztat%C3%A1sok/Norm%C3%A1lt_t%C3%A9r" rel="nofollow" title="Az erről a lapról hivatkozott lapok utolsó változtatásai [k]" accesskey="k"><span>Kapcsolódó változtatások</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Speci%C3%A1lis_lapok" title="Az összes speciális lap listája [q]" accesskey="q"><span>Speciális lapok</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&oldid=27546338" title="Állandó hivatkozás ezen lap ezen változatához"><span>Hivatkozás erre a változatra</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=info" title="További információk erről a lapról"><span>Lapinformációk</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Hivatkoz%C3%A1s&page=Norm%C3%A1lt_t%C3%A9r&id=27546338&wpFormIdentifier=titleform" title="Információk a lap idézésével kapcsolatban"><span>Hogyan hivatkozz erre a lapra?</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:UrlShortener&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FNorm%25C3%25A1lt_t%25C3%25A9r"><span>Rövidített URL készítése</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:QrCode&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FNorm%25C3%25A1lt_t%25C3%25A9r"><span>QR-kód letöltése</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Nyomtatás/exportálás </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:K%C3%B6nyv&bookcmd=book_creator&referer=Norm%C3%A1lt+t%C3%A9r"><span>Könyv készítése</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:DownloadAsPdf&page=Norm%C3%A1lt_t%C3%A9r&action=show-download-screen"><span>Letöltés PDF-ként</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&printable=yes" title="A lap nyomtatható változata [p]" accesskey="p"><span>Nyomtatható változat</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Társprojektek </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Normed_spaces" hreflang="en"><span>Wikimédia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q726210" title="Kapcsolt adattárelem [g]" accesskey="g"><span>Wikidata-adatlap</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Megjelenés"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Megjelenés</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">elrejtés</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 id="mw-indicator-indicator-fr-review-status" class="mw-indicator"><indicator name="fr-review-status" class="mw-fr-review-status-indicator" id="mw-fr-revision-toggle"><span class="cdx-fr-css-icon-review--status--stable"></span><b>Ellenőrzött</b></indicator></div> </div> <div id="siteSub" class="noprint">A Wikipédiából, a szabad enciklopédiából</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div id="mw-fr-revision-messages"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">Változat állapota</h2><p class="cdx-dialog__header__subtitle">Ez a lap egy ellenőrzött változata</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Ez a <a href="/wiki/Wikip%C3%A9dia:Jel%C3%B6lt_lapv%C3%A1ltozatok" title="Wikipédia:Jelölt lapváltozatok">közzétett változat</a>, <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Rendszernapl%C3%B3k&type=review&page=Norm%C3%A1lt_t%C3%A9r">ellenőrizve</a>: <i>2024. október 26.</i><p><table id="mw-fr-revisionratings-box" class="flaggedrevs-color-1" style="margin: auto;" cellpadding="0"><tr><td class="fr-text" style="vertical-align: middle;">Pontosság</td><td class="fr-value40" style="vertical-align: middle;">ellenőrzött</td></tr></table></p></div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hu" dir="ltr"><p>A <b>normált tér</b> <a href="/wiki/Matematika" title="Matematika">matematikai</a> objektum, a <a href="/wiki/Line%C3%A1ris_algebra" title="Lineáris algebra">lineáris algebra</a>, az <a href="/wiki/Matematikai_anal%C3%ADzis" title="Matematikai analízis">analízis</a> és azon belül a <a href="/wiki/Funkcion%C3%A1lanal%C3%ADzis" title="Funkcionálanalízis">funkcionálanalízis</a> vizsgálja. A <a href="/wiki/Parci%C3%A1lis_differenci%C3%A1legyenlet" title="Parciális differenciálegyenlet">parciális differenciálegyenletek</a> és az <a href="/w/index.php?title=Integr%C3%A1legyenlet&action=edit&redlink=1" class="new" title="Integrálegyenlet (a lap nem létezik)">integrálegyenletek</a> megoldásához használják. Fontos speciális esete a közönséges 3 <a href="/wiki/Dimenzi%C3%B3" title="Dimenzió">dimenziós</a> tér. Valójában a normált tér éppen ennek egy természetes általánosítása. A normált tér vektortér, amin <a href="/wiki/Norma_(matematika)" title="Norma (matematika)">norma</a> is értelmezve van, azaz a vektoroknak van hosszuk. A norma <a href="/w/index.php?title=Metrika&action=edit&redlink=1" class="new" title="Metrika (a lap nem létezik)">metrikát</a> indukált, ezzel a normált tér <a href="/wiki/Metrikus_t%C3%A9r" title="Metrikus tér">metrikus tér</a>. A metrika topológiát indukál, ezzel a tér <a href="/wiki/Topologikus_t%C3%A9r" title="Topologikus tér">topologikus tér</a>. Ha egy normált tér teljes, akkor teljes metrikus tér, avagy <a href="/wiki/Banach-t%C3%A9r" title="Banach-tér">Banach-tér</a>. Normált tér származhat <a href="/wiki/Skal%C3%A1rszorzatos_vektort%C3%A9r" title="Skalárszorzatos vektortér">skalárszorzatos vektortérből</a> vagy <a href="/w/index.php?title=F%C3%A9lnorm%C3%A1lt_t%C3%A9r&action=edit&redlink=1" class="new" title="Félnormált tér (a lap nem létezik)">félnormált térből</a> <a href="/w/index.php?title=Faktort%C3%A9r&action=edit&redlink=1" class="new" title="Faktortér (a lap nem létezik)">faktortérként</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definíció"><span id="Defin.C3.ADci.C3.B3"></span>Definíció</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=1" title="Szakasz szerkesztése: Definíció"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <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 (V,||\cdot ||)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </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"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,||\cdot ||)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650a2d1c77e7b16f7119a451eda0b5ecadfd16bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.897ex; height:2.843ex;" alt="{\displaystyle (V,||\cdot ||)}"></span> kettőst <b>normált térnek</b> nevezzük, ha <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 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></span><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}"></span> <a href="/wiki/Vektort%C3%A9r" title="Vektortér">vektortér</a> a <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 {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"></span> <a href="/wiki/Test_(algebra)" title="Test (algebra)">számtest</a> felett, ahol <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 {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"></span> a <a href="/wiki/Komplex_sz%C3%A1mok" title="Komplex számok">komplex</a> vagy <a href="/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok">valós számok</a> <a href="/wiki/Test_(algebra)" title="Test (algebra)">teste</a>, a <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 ||\cdot ||:V\to \mathbb {R} }"> <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"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mi>V</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 ||\cdot ||:V\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739039e9cdb72c5195b3541b91dd8583bd57d94b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.283ex; height:2.843ex;" alt="{\displaystyle ||\cdot ||:V\to \mathbb {R} }"></span> <a href="/wiki/F%C3%BCggv%C3%A9ny_(matematika)" title="Függvény (matematika)">függvény</a> pedig egy <a href="/wiki/Norma_(matematika)" title="Norma (matematika)">norma</a>, amelyre teljesülnek az alábbi tulajdonságok: </p> <ol><li><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 \forall {\vec {x}}\in V\ ||{\vec {x}}||\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>V</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {\vec {x}}\in V\ ||{\vec {x}}||\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee298e79691241660a0964ccd1cff394889024b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.009ex; height:2.843ex;" alt="{\displaystyle \forall {\vec {x}}\in V\ ||{\vec {x}}||\geq 0}"></span> (szemidefinitség)</li> <li><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 ||{\vec {x}}||=0\Leftrightarrow {\vec {x}}={\vec {0}}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||{\vec {x}}||=0\Leftrightarrow {\vec {x}}={\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f0085fb89190679c9d73e55ee4dde86dc7c301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.383ex; height:3.343ex;" alt="{\displaystyle ||{\vec {x}}||=0\Leftrightarrow {\vec {x}}={\vec {0}}}"></span> (definitség)</li> <li><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 \forall \alpha \in \mathbb {K} \ \forall {\vec {x}}\in V\ ||\alpha \cdot {\vec {x}}||=|\alpha |\cdot ||{\vec {x}}||}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>V</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \alpha \in \mathbb {K} \ \forall {\vec {x}}\in V\ ||\alpha \cdot {\vec {x}}||=|\alpha |\cdot ||{\vec {x}}||}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b165a32d863c0c5064686263ed8960966d00ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.4ex; height:2.843ex;" alt="{\displaystyle \forall \alpha \in \mathbb {K} \ \forall {\vec {x}}\in V\ ||\alpha \cdot {\vec {x}}||=|\alpha |\cdot ||{\vec {x}}||}"></span> (abszolút homogenitás)</li> <li><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 ||{\vec {x}}+{\vec {y}}||\leq ||{\vec {x}}||+||{\vec {y}}||}"> <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||{\vec {x}}+{\vec {y}}||\leq ||{\vec {x}}||+||{\vec {y}}||}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b9775490dfb2315ba11f6bea5a569861db6072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.73ex; height:2.843ex;" alt="{\displaystyle ||{\vec {x}}+{\vec {y}}||\leq ||{\vec {x}}||+||{\vec {y}}||}"></span> (szubadditivitás, háromszög-egyenlőtlenség)</li></ol> <div class="mw-heading mw-heading2"><h2 id="Példák"><span id="P.C3.A9ld.C3.A1k"></span>Példák</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=2" title="Szakasz szerkesztése: Példák"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Legegyszerűbb példák a véges dimenziós <a href="/wiki/Komplex_sz%C3%A1mok" title="Komplex számok">komplex</a> és <a href="/wiki/Val%C3%B3s_sz%C3%A1mok" title="Valós számok">valós</a> vektorterek, rajtuk az úgynevezett <a href="/wiki/Euklideszi_norma" title="Euklideszi norma">euklideszi normával</a>. Ha <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 {\vec {x}}\in \mathbb {K} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}\in \mathbb {K} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1b2783dde019e4b2bea0131fc7ddb10410815e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.197ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}\in \mathbb {K} ^{n}}"></span>, akkor ennek euklideszi normája: </p> <dl><dd><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 ||{\vec {x}}||_{E}={\sqrt {|x_{1}|^{2}+|x_{2}|^{2}+\ldots +|x_{n}|^{2}}}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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> <mo>…</mo> <mo>+</mo> <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> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||{\vec {x}}||_{E}={\sqrt {|x_{1}|^{2}+|x_{2}|^{2}+\ldots +|x_{n}|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444fbf5fc176d6ff18985cf8779e07fd2b903d5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:36.431ex; height:4.843ex;" alt="{\displaystyle ||{\vec {x}}||_{E}={\sqrt {|x_{1}|^{2}+|x_{2}|^{2}+\ldots +|x_{n}|^{2}}}}"></span></dd></dl> <p>Más normák is értelmezhetőek ezen a vektortéren: </p> <dl><dd><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 ||{\vec {x}}||_{\max }=\max\{|x_{1}|,|x_{2}|,\ldots ,|x_{n}|\}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||{\vec {x}}||_{\max }=\max\{|x_{1}|,|x_{2}|,\ldots ,|x_{n}|\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c1b40d04a5210f78b9ab8bb94afabf21a03ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.367ex; height:3.009ex;" alt="{\displaystyle ||{\vec {x}}||_{\max }=\max\{|x_{1}|,|x_{2}|,\ldots ,|x_{n}|\}}"></span></dd> <dd><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 ||{\vec {x}}||_{p}={\sqrt[{p}]{|x_{1}|^{p}+|x_{2}|^{p}+\ldots +|x_{n}|^{p}}}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <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"> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||{\vec {x}}||_{p}={\sqrt[{p}]{|x_{1}|^{p}+|x_{2}|^{p}+\ldots +|x_{n}|^{p}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209fcb12f46d593b70493f1cc2065e2d95e1435b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.017ex; height:4.843ex;" alt="{\displaystyle ||{\vec {x}}||_{p}={\sqrt[{p}]{|x_{1}|^{p}+|x_{2}|^{p}+\ldots +|x_{n}|^{p}}}}"></span></dd></dl> <p>Ha adott két normált tér, akkor egy köztük ható <a href="/wiki/Line%C3%A1ris_lek%C3%A9pez%C3%A9s" title="Lineáris leképezés">lineáris operátor</a> normáját is lehet értelmezni. Legyen ugyanis <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,||\cdot ||_{X}),\ (Y,||\cdot ||_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,||\cdot ||_{X}),\ (Y,||\cdot ||_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a91ef2b42e5b6ce6b647e29bb38ece74dd33c6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.706ex; height:2.843ex;" alt="{\displaystyle (X,||\cdot ||_{X}),\ (Y,||\cdot ||_{Y})}"></span> két normált tér, <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 {\mathcal {A}}:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61bfbd73105e2e4ed9571177b932c574469d333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.208ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}:X\to Y}"></span> egy lineáris operátor. Ennek <a href="/wiki/Oper%C3%A1tornorma" title="Operátornorma">(operátor)normája</a>: </p> <dl><dd><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 ||{\mathcal {A}}||=\sup\{||{\mathcal {A}}({\vec {x}})||_{Y}:||{\vec {x}}||_{X}\leq 1\}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||{\mathcal {A}}||=\sup\{||{\mathcal {A}}({\vec {x}})||_{Y}:||{\vec {x}}||_{X}\leq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4259104c6c9b0a90a7d10fe0743847a8449af934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.279ex; height:2.843ex;" alt="{\displaystyle ||{\mathcal {A}}||=\sup\{||{\mathcal {A}}({\vec {x}})||_{Y}:||{\vec {x}}||_{X}\leq 1\}}"></span>, feltéve hogy ez a <a href="/w/index.php?title=Szupr%C3%A9mum&action=edit&redlink=1" class="new" title="Szuprémum (a lap nem létezik)">szuprémum</a> véges.</dd></dl> <p>Az olyan lineáris operátorokat, amelyekre ez véges, <b>korlátos lineáris operátoroknak</b> nevezzük. Jegyezzük meg, hogy ezek az operátorok pontosan a <a href="/wiki/Folytonos_f%C3%BCggv%C3%A9ny" title="Folytonos függvény">folytonos</a> lineáris operátorok! </p><p><a href="/wiki/F%C3%BCggv%C3%A9nyt%C3%A9r" title="Függvénytér">Függvénytereken</a> is lehet normát értelmezni. Legyen <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,{\mathcal {A}},\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {A}},\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d634d210e57700027029694595ffea10410bf0d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {A}},\mu )}"></span> <a href="/wiki/M%C3%A9rt%C3%A9kt%C3%A9r" title="Mértéktér">mértéktér</a> (vigyázat, az <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"></span> itt már egy <a href="/wiki/%CE%A3-algebra" title="Σ-algebra">σ-algebra</a>), és vegyük a következő függvényteret: </p> <dl><dd><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 L^{p}(X)=\{f:X\to \mathbb {K} :\int _{X}|f|^{p}d\mu <\infty \}}"> <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>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>:</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mi>d</mi> <mi>μ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(X)=\{f:X\to \mathbb {K} :\int _{X}|f|^{p}d\mu <\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2191135d0ec70470b3435f1946352c768354d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.393ex; height:5.676ex;" alt="{\displaystyle L^{p}(X)=\{f:X\to \mathbb {K} :\int _{X}|f|^{p}d\mu <\infty \}}"></span></dd></dl> <p>Vezessünk be ezen egy <a href="/wiki/Ekvivalenciarel%C3%A1ci%C3%B3" title="Ekvivalenciareláció">ekvivalencia-relációt</a>: </p> <dl><dd><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 f\sim g\Leftrightarrow \mu \left(\{x:f(x)\not =g(x)\}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∼</mo> <mi>g</mi> <mo stretchy="false">⇔</mo> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\sim g\Leftrightarrow \mu \left(\{x:f(x)\not =g(x)\}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa307dcc5cfab61da4ec766722c7ef08767793ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.329ex; height:2.843ex;" alt="{\displaystyle f\sim g\Leftrightarrow \mu \left(\{x:f(x)\not =g(x)\}\right)=0}"></span></dd></dl> <p>Az ekvivalenciaosztályokat egy reprezentánsukkal szokás jelölni, míg a relációval faktorizált <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 L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"></span>-t szintén <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 L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"></span>-vel. </p><p>Legyen most <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 f\in L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886e75c9dd0ab24fb24d63c312fb04110a9b0f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.761ex; height:2.676ex;" alt="{\displaystyle f\in L^{p}}"></span>, és ekkor </p> <dl><dd><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 ||f||_{p}=\left(\int _{X}|f|^{p}d\mu \right)^{1/p}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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>X</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mi>d</mi> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||f||_{p}=\left(\int _{X}|f|^{p}d\mu \right)^{1/p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc1e9e8e727ca0b22c48b03f35870de067859f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.709ex; height:6.676ex;" alt="{\displaystyle ||f||_{p}=\left(\int _{X}|f|^{p}d\mu \right)^{1/p}}"></span>.</dd></dl> <p>Ennek valójában speciális esete a következő: </p> <dl><dd><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 f\in C([a,b])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in C([a,b])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35a84e8df64b276d8617c4ce40b9a2dd5bfe6a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.25ex; height:2.843ex;" alt="{\displaystyle f\in C([a,b])}"></span> esetén <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 ||f||_{\infty }=\sup\{f(x):x\in [a,b]\}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||f||_{\infty }=\sup\{f(x):x\in [a,b]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c22a58188b525338ef8994d8550460c22327a647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.746ex; height:3.009ex;" alt="{\displaystyle ||f||_{\infty }=\sup\{f(x):x\in [a,b]\}}"></span>.</dd></dl> <p>További példák: </p> <ul><li>A valós vagy komplex számok az abszolútértékkel, mint normával: <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 {K} },|\cdot |)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbb {K} },|\cdot |)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9a9c9b85617f47da6059d85ab6a939d7c34726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.624ex; height:2.843ex;" alt="{\displaystyle ({\mathbb {K} },|\cdot |)}"></span></li> <li>Valós vagy komplex <a href="/wiki/M%C3%A1trix_(matematika)" title="Mátrix (matematika)">mátrixok</a> a <a href="/wiki/Frobenius-norma" title="Frobenius-norma">Frobenius-normával</a>: <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 {K} }^{m,n},\|\cdot \|_{F})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbb {K} }^{m,n},\|\cdot \|_{F})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5ab03409a526fe6d9e8e7af0e1e8a3a43d28be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.237ex; height:2.843ex;" alt="{\displaystyle ({\mathbb {K} }^{m,n},\|\cdot \|_{F})}"></span></li> <li>Valós vagy komplex értékű korlátos függvények a szuprémumnormával: <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 (B,\|\cdot \|_{\sup })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">sup</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B,\|\cdot \|_{\sup })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90de71a8fdafd13b20233f1073988e660d25daef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.319ex; height:3.009ex;" alt="{\displaystyle (B,\|\cdot \|_{\sup })}"></span></li> <li>Kompakt halmazon értelmezett valós vagy komplex értékű folytonos függvények a maximumnormával: <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 (C^{0},\|\cdot \|_{\max })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C^{0},\|\cdot \|_{\max })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d27182bcb124b7791982af50fa282889895208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.99ex; height:3.176ex;" alt="{\displaystyle (C^{0},\|\cdot \|_{\max })}"></span></li> <li>Az <i>m</i>-szer <a href="/w/index.php?title=Folytonos_differenci%C3%A1lhat%C3%B3s%C3%A1g&action=edit&redlink=1" class="new" title="Folytonos differenciálhatóság (a lap nem létezik)">folytonosan differenciálható</a> valós vagy komplex értékű <a href="/w/index.php?title=Korl%C3%A1tos_f%C3%BCggv%C3%A9ny&action=edit&redlink=1" class="new" title="Korlátos függvény (a lap nem létezik)">korlátos függvények</a> a <i>C<sup>m</sup></i>-normával: <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 (C^{m},\|\cdot \|_{C^{m}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C^{m},\|\cdot \|_{C^{m}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb630e358389fcb435dc25de22b23c76ab87d5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.159ex; height:2.843ex;" alt="{\displaystyle (C^{m},\|\cdot \|_{C^{m}})}"></span></li> <li>Két valós vagy komplex vektortér között menő korlátos lineáris operátorok tere az <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 ({\mathfrak {L}}(V,W),\|\cdot \|_{{\mathfrak {L}}(V,W)})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">L</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathfrak {L}}(V,W),\|\cdot \|_{{\mathfrak {L}}(V,W)})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6078e300a2eb8a450925643acf3d0473859ccccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.51ex; height:3.176ex;" alt="{\displaystyle ({\mathfrak {L}}(V,W),\|\cdot \|_{{\mathfrak {L}}(V,W)})}"></span> operátornormával</li></ul> <div class="mw-heading mw-heading2"><h2 id="Tulajdonságok"><span id="Tulajdons.C3.A1gok"></span>Tulajdonságok</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=3" title="Szakasz szerkesztése: Tulajdonságok"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Kapcsolat_a_metrikus_terekkel">Kapcsolat a metrikus terekkel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=4" title="Szakasz szerkesztése: Kapcsolat a metrikus terekkel"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Minden <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 (V,||\cdot ||)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </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"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,||\cdot ||)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/650a2d1c77e7b16f7119a451eda0b5ecadfd16bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.897ex; height:2.843ex;" alt="{\displaystyle (V,||\cdot ||)}"></span> normált tér <a href="/wiki/Metrikus_t%C3%A9r" title="Metrikus tér">metrizálható</a>. Ha ugyanis <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 {\vec {x}},{\vec {y}}\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}},{\vec {y}}\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a520cbc346b98e92f4414ad770a3dd038aef36b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.256ex; height:2.676ex;" alt="{\displaystyle {\vec {x}},{\vec {y}}\in V}"></span>, akkor ezek távolságát, <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 \varrho ({\vec {x}},{\vec {y}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϱ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varrho ({\vec {x}},{\vec {y}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9365ec6593c2be06be0c33a453f495d97594c34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.639ex; height:2.843ex;" alt="{\displaystyle \varrho ({\vec {x}},{\vec {y}})}"></span>-t definiálhatjuk a következőképp: </p> <dl><dd><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 \varrho ({\vec {x}},{\vec {y}})=||{\vec {x}}-{\vec {y}}||}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϱ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varrho ({\vec {x}},{\vec {y}})=||{\vec {x}}-{\vec {y}}||}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574336c90204d8051ed220e00be17854d2cc422f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.76ex; height:2.843ex;" alt="{\displaystyle \varrho ({\vec {x}},{\vec {y}})=||{\vec {x}}-{\vec {y}}||}"></span></dd></dl> <p>Ezzel egyben azt is látjuk, hogy a norma segítségével <a href="/wiki/Topologikus_t%C3%A9r" title="Topologikus tér">topológiát</a> definiálhatunk, így van értelme már fentebb említett folytonosságról beszélni normált terek között. Fontos megjegyezni, hogy egyazon vektortéren két különböző norma nem feltétlen ad <a href="/wiki/Homeomorfia" title="Homeomorfia">homeomorf</a> topologikus struktúrát. </p> <div class="mw-heading mw-heading3"><h3 id="Ekvivalens_normák"><span id="Ekvivalens_norm.C3.A1k"></span>Ekvivalens normák</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=5" title="Szakasz szerkesztése: Ekvivalens normák"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Legyen adva <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 (V,||\cdot ||_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,||\cdot ||_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768e7095bb15c44257ff3f47d753b7dba64c876e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.951ex; height:2.843ex;" alt="{\displaystyle (V,||\cdot ||_{1})}"></span> és <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 (V,||\cdot ||_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,||\cdot ||_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df9d6a0e9131ed9d716aa40fb506c6c916452b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.951ex; height:2.843ex;" alt="{\displaystyle (V,||\cdot ||_{2})}"></span>, azaz egyazon vektortéren két különböző norma. Azt mondjuk, hogy ők <b>ekvivalensek</b>, ha létezik olyan <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 0<a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5963861dca1837680ebf05e63b8e21db9748a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.041ex; height:2.509ex;" alt="{\displaystyle 0<a,b\in \mathbb {R} }"></span>, hogy minden <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 {\vec {x}}\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8c20331b5ded219ce471a0a8ca64a25ae89a16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}\in V}"></span> esetén: </p> <dl><dd><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 a\leq {\frac {||{\vec {x}}||_{1}}{||{\vec {x}}||_{2}}}\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq {\frac {||{\vec {x}}||_{1}}{||{\vec {x}}||_{2}}}\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198dd9e69c92ce0e86623884289f3cbc2f9cb808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.232ex; height:6.509ex;" alt="{\displaystyle a\leq {\frac {||{\vec {x}}||_{1}}{||{\vec {x}}||_{2}}}\leq b}"></span></dd></dl> <p>Ekkor <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 (V,||\cdot ||_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,||\cdot ||_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768e7095bb15c44257ff3f47d753b7dba64c876e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.951ex; height:2.843ex;" alt="{\displaystyle (V,||\cdot ||_{1})}"></span> és <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 (V,||\cdot ||_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,||\cdot ||_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df9d6a0e9131ed9d716aa40fb506c6c916452b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.951ex; height:2.843ex;" alt="{\displaystyle (V,||\cdot ||_{2})}"></span> homeomorfak, ugyanis az <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 \operatorname {Id} :V\to V,\ \operatorname {Id} ({\vec {x}})={\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Id</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> <mo>,</mo> <mtext> </mtext> <mi>Id</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Id} :V\to V,\ \operatorname {Id} ({\vec {x}})={\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61fc7711163306544a0165c3c060282cc026ad12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.959ex; height:2.843ex;" alt="{\displaystyle \operatorname {Id} :V\to V,\ \operatorname {Id} ({\vec {x}})={\vec {x}}}"></span> <a href="/wiki/F%C3%BCggv%C3%A9ny_(matematika)" title="Függvény (matematika)">függvény</a> az <a href="/wiki/Inverz_f%C3%BCggv%C3%A9ny" title="Inverz függvény">inverzével</a> együtt teljesíti a <a href="/wiki/Lipschitz-tulajdons%C3%A1g" title="Lipschitz-tulajdonság">Lipschitz-feltételt</a>. </p><p>Bizonyítható, hogy egy (valós vagy komplex) vektortér akkor és csak akkor véges dimenziós, ha tetszőleges két rajta értelmezett norma ekvivalens. </p> <div class="mw-heading mw-heading3"><h3 id="Normált_terek_szorzata"><span id="Norm.C3.A1lt_terek_szorzata"></span>Normált terek szorzata</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=6" title="Szakasz szerkesztése: Normált terek szorzata"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Legyen <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 (V,||\cdot ||_{V})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,||\cdot ||_{V})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07aebc2d88e32d6750693ea78ec7e23c2e516a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.393ex; height:3.009ex;" alt="{\displaystyle (V,||\cdot ||_{V})}"></span> és <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 (W,||\cdot ||_{W})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>W</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (W,||\cdot ||_{W})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b7b2c24c4971b6063ebacac98f8430622c4a7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.499ex; height:3.009ex;" alt="{\displaystyle (W,||\cdot ||_{W})}"></span> két normált tér. A <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 V\times W=\{({\vec {v}},{\vec {w}}):{\vec {v}}\in V,{\vec {w}}\in W\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>×</mo> <mi>W</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>W</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\times W=\{({\vec {v}},{\vec {w}}):{\vec {v}}\in V,{\vec {w}}\in W\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f78711f4cf537d99bab0c687ff20ba8231e5bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.883ex; height:2.843ex;" alt="{\displaystyle V\times W=\{({\vec {v}},{\vec {w}}):{\vec {v}}\in V,{\vec {w}}\in W\}}"></span> vektortéren szintén értelmezhető normált tér struktúra: </p> <dl><dd><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 ||({\vec {v}},{\vec {w}})||_{1}=\max\{||{\vec {v}}||_{V},||{\vec {w}}||_{W}\}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||({\vec {v}},{\vec {w}})||_{1}=\max\{||{\vec {v}}||_{V},||{\vec {w}}||_{W}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffe11b6896b388dda534fbcc89e8dc7170479fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.572ex; height:3.009ex;" alt="{\displaystyle ||({\vec {v}},{\vec {w}})||_{1}=\max\{||{\vec {v}}||_{V},||{\vec {w}}||_{W}\}}"></span></dd> <dd><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 ||({\vec {v}},{\vec {w}})||_{2}=||{\vec {v}}||_{V}+||{\vec {w}}||_{W}}"> <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"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||({\vec {v}},{\vec {w}})||_{2}=||{\vec {v}}||_{V}+||{\vec {w}}||_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a58473e34e108cc96a84c940c9d2cc4a3bd4ccc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.728ex; height:3.009ex;" alt="{\displaystyle ||({\vec {v}},{\vec {w}})||_{2}=||{\vec {v}}||_{V}+||{\vec {w}}||_{W}}"></span></dd></dl> <p>Megmutatható, hogy a fenti két norma ekvivalens. </p> <div class="mw-heading mw-heading2"><h2 id="Speciális_esetek"><span id="Speci.C3.A1lis_esetek"></span>Speciális esetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=7" title="Szakasz szerkesztése: Speciális esetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Skalárszorzatos_vektorterek"><span id="Skal.C3.A1rszorzatos_vektorterek"></span>Skalárszorzatos vektorterek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=8" title="Szakasz szerkesztése: Skalárszorzatos vektorterek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Egy norma származhat skalárszorzatból. Minden <a href="/wiki/Skal%C3%A1rszorzatos_vektort%C3%A9r" title="Skalárszorzatos vektortér">skalárszorzatos vektortér</a> normált tér a <a href="/wiki/Skal%C3%A1rszorzat" class="mw-redirect" title="Skalárszorzat">skalárszorzat</a> által indukált normával: </p> <dl><dd><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\|={\sqrt {\langle x,x\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a885f02102ba1179708035546be39be936979b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.579ex; height:4.843ex;" alt="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}"></span></dd></dl> <p>Egy norma pontosan akkor származik skalárszorzatból, ha teljesíti a <a href="/wiki/Paralelogrammaazonoss%C3%A1g" title="Paralelogrammaazonosság">paralelogrammaazonosságot</a>. A teljes skalárszorzatos vektorterek <a href="/wiki/Hilbert-t%C3%A9r" title="Hilbert-tér">Hilbert-terek</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Teljes_terek">Teljes terek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=9" title="Szakasz szerkesztése: Teljes terek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Egy normált tér teljes, ha a térben minden <a href="/wiki/Cauchy-sorozat" title="Cauchy-sorozat">Cauchy-sorozat</a> <a href="/wiki/Konvergencia_(matematika)" title="Konvergencia (matematika)">konvergens</a>. Egy teljes normált tér <a href="/wiki/Banach-t%C3%A9r" title="Banach-tér">Banach-tér</a>. Minden normált tér teljessé tehető Cauchy-sorozatok ekvivalenciaosztályaival. Így Banach-teret kapunk, ami az eredeti teret <a href="/wiki/S%C5%B1r%C5%B1_r%C3%A9szhalmaz" title="Sűrű részhalmaz">sűrű</a> altérként tartalmazza. </p> <div class="mw-heading mw-heading2"><h2 id="Általánosítások"><span id=".C3.81ltal.C3.A1nos.C3.ADt.C3.A1sok"></span>Általánosítások</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=10" title="Szakasz szerkesztése: Általánosítások"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Félnormált_terek"><span id="F.C3.A9lnorm.C3.A1lt_terek"></span>Félnormált terek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=11" title="Szakasz szerkesztése: Félnormált terek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ha <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 \|\cdot \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"></span> félnorma, akkor <a href="/w/index.php?title=F%C3%A9lnorm%C3%A1lt_t%C3%A9r&action=edit&redlink=1" class="new" title="Félnormált tér (a lap nem létezik)">félnormált térről</a> van szó. Egy félnormált térből normált tér kapható faktortérképzéssel. Az <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}"> <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></span><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}"></span> és <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 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></span><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}"></span> vektorok akkor tartoznak egy ekvivalenciaosztályba, ha <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-y\|=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x-y\|=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b23da6e1d8439f6159ad0691ea4e7d1c2544673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.911ex; height:2.843ex;" alt="{\displaystyle \|x-y\|=0}"></span>. A funkcionálanalízisben a normált terek mellett <a href="/w/index.php?title=Lok%C3%A1lisan_konvex_t%C3%A9r&action=edit&redlink=1" class="new" title="Lokálisan konvex tér (a lap nem létezik)">lokálisan konvex terekkel</a> is foglalkoznak, melyek félnormák egy halmazával ellátott vektorterek. </p> <div class="mw-heading mw-heading3"><h3 id="Metrikus_és_topologikus_terek"><span id="Metrikus_.C3.A9s_topologikus_terek"></span>Metrikus és topologikus terek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=12" title="Szakasz szerkesztése: Metrikus és topologikus terek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Minden norma <a href="/w/index.php?title=Induk%C3%A1lt_metrika&action=edit&redlink=1" class="new" title="Indukált metrika (a lap nem létezik)">metrikát indukál</a>: </p> <dl><dd><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(x,y)=\|x-y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=\|x-y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94df456fedac6b08c33cb4dffa5345a0ce0891f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.293ex; height:2.843ex;" alt="{\displaystyle d(x,y)=\|x-y\|}"></span></dd></dl> <p>Minden normált tér <a href="/wiki/Metrikus_t%C3%A9r" title="Metrikus tér">metrikus tér</a> a <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 (V,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a88b6a2de93702959fc67450e18ab8b0f67f48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.846ex; height:2.843ex;" alt="{\displaystyle (V,d)}"></span> metrikával, sőt, <a href="/wiki/Topologikus_t%C3%A9r" title="Topologikus tér">topologikus tér</a> a norma által indukált topológiára: <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 (V,{\mathcal {T}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V,{\mathcal {T}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c58960078f3b3bde1263b7dd931a8e4eac3e6980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.566ex; height:2.843ex;" alt="{\displaystyle (V,{\mathcal {T}})}"></span>. Így értelmezhetők normált terekben topológiai fogalmak, mint <a href="/wiki/Hat%C3%A1r%C3%A9rt%C3%A9k" title="Határérték">határérték</a>, <a href="/wiki/Cauchy-sorozat" title="Cauchy-sorozat">Cauchy-sorozat</a>, <a href="/w/index.php?title=Folytonoss%C3%A1g_(topol%C3%B3gia)&action=edit&redlink=1" class="new" title="Folytonosság (topológia) (a lap nem létezik)">folytonosság</a> és <a href="/wiki/Kompakts%C3%A1g" title="Kompaktság">kompaktság</a>. Így egy <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_{n})_{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"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db535bd8b18bb7f5105d670ad27e611d1fa00694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.576ex; height:2.843ex;" alt="{\displaystyle (x_{n})_{n}}"></span> pontosan akkor tart az <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}"> <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></span><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}"></span> határértékhez, ha <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_{n}-x\|\rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x_{n}-x\|\rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0cdd7a1869c4135900847a14f6ab8460606d40f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.82ex; height:2.843ex;" alt="{\displaystyle \|x_{n}-x\|\rightarrow 0}"></span>. Maga a norma is folytonos az általa indukált topológiában. </p><p>A metrikus tér valódi általánosítása a normált térnek, ugyanis vannak <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,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4d7a16bca9e216c0221b43a1c3377aa5e358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.843ex;" alt="{\displaystyle (X,d)}"></span> metrikus terek, ahol: </p> <ul><li>a <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> metrika nem ábrázolható normával</li> <li><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}"> <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></span><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}"></span>-ben nincs vektortér struktúra értelmezve.</li></ul> <p>Ekvivalens normák ugyanazt az <a href="/wiki/Uniform_t%C3%A9r" title="Uniform tér">uniform struktúrát</a> indukálják, és ezzel ugyanazt a topológiát is. Véges dimenziós vektorterekben minden norma ekvivalens; végtelen dimenzióban azonban ez nincs így. </p><p>Egy topologikus vektortér normálható, ha topológiája normával indukálható. Kolmogorov normálhatósági kritériuma szerint egy topologikus <a href="/wiki/Hausdorff-t%C3%A9r" title="Hausdorff-tér">Hausdorff-tér</a> topológiája pontosan akkor indukálható normával, ha nullvektorának van konvex korlátos környezete. </p> <div class="mw-heading mw-heading3"><h3 id="Abszolútértékes_testek"><span id="Abszol.C3.BAt.C3.A9rt.C3.A9kes_testek"></span>Abszolútértékes testek</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=13" title="Szakasz szerkesztése: Abszolútértékes testek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A normált tér fogalma általánosítható, ha a valós vagy komplex test helyett általánosabban egy <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 (K,|\cdot |)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,|\cdot |)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e460caabf94248d3ca86b2dd778bf0c414a43a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.882ex; height:2.843ex;" alt="{\displaystyle (K,|\cdot |)}"></span> testet veszünk, ahol <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 |\cdot |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4570d0a1c9fb8f2f413f0b73ce846dd1eb1dca3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.973ex; height:2.843ex;" alt="{\displaystyle |\cdot |}"></span> abszolútérték.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Jegyzetek">Jegyzetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=14" title="Szakasz szerkesztése: Jegyzetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="book citation" style="font-style:normal">Falko Lorenz. <i>Einführung in die Algebra II</i>, 2., Spektrum Akademischer Verlag, 69. o. (1997)</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einf%C3%BChrung+in+die+Algebra+II&rft.au=Falko+Lorenz&rft.date=1997&rft.edition=2.&rft.pub=Spektrum+Akademischer+Verlag&rft.pages=69"><span style="display: none;"> </span></span></span> </li> </ol></div></div><div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"></div></div> <div class="mw-heading mw-heading2"><h2 id="Források"><span id="Forr.C3.A1sok"></span>Források</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=15" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Robert E. Megginson: <i>An Introduction to Banach Space Theory.</i> Springer-Verlag, 1998, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/0387984313" title="Speciális:Könyvforrások/0387984313">ISBN 0-387-98431-3</a></li> <li><a href="/w/index.php?title=Dirk_Werner_(Mathematiker)&action=edit&redlink=1" class="new" title="Dirk Werner (Mathematiker) (a lap nem létezik)">Dirk Werner</a>: <i>Funktionalanalysis.</i> 5., erweiterte Auflage. Springer, Berlin u. a. 2005, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/3540435867" title="Speciális:Könyvforrások/3540435867">ISBN 3-540-43586-7</a>, Kapitel I.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Fordítás"><span id="Ford.C3.ADt.C3.A1s"></span>Fordítás</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Norm%C3%A1lt_t%C3%A9r&action=edit&section=16" title="Szakasz szerkesztése: Fordítás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ez a szócikk részben vagy egészben a <i><a href="https://de.wikipedia.org/wiki/Normierter_Raum" class="extiw" title="de:Normierter Raum">Normierter Raum</a></i> című német Wikipédia-szócikk fordításán alapul. Az eredeti cikk szerkesztőit annak laptörténete sorolja fel. 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