Hilbert-tér – Wikipédia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="hu" dir="ltr"> <head> <meta charset="UTF-8"> <title>Hilbert-tér – Wikipédia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )huwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"ymd","wgMonthNames":["","január","február","március","április","május","június","július","augusztus","szeptember","október","november","december"],"wgRequestId":"91903d6c-0418-4e62-bb01-bc5b3a9147ac","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Hilbert-tér","wgTitle":"Hilbert-tér","wgCurRevisionId":27407869,"wgRevisionId":27407869,"wgArticleId":38923,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Wikipédia-szócikkek LCCN-azonosítóval","Wikipédia-szócikkek GND-azonosítóval","Wikipédia-szócikkek BNF-azonosítóval","Wikipédia-szócikkek KKT-azonosítóval","Lapok elavult formátumú matematikai tagekkel","Analízis"],"wgPageViewLanguage":"hu","wgPageContentLanguage":"hu","wgPageContentModel":"wikitext","wgRelevantPageName":"Hilbert-tér","wgRelevantArticleId":38923,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable": true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":2}}},"wgStableRevisionId":27407869,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"hu","pageLanguageDir":"ltr","pageVariantFallbacks":"hu"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q190056","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false, "wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.infobox":"ready","ext.gadget.wikiMenuStyles":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.wdsearch","ext.gadget.irclogin", "ext.gadget.ImageAnnotator.loader","ext.gadget.collapsible","ext.gadget.kepdia","ext.gadget.kinai","ext.gadget.poziciosTerkep","ext.gadget.wikiMenu","ext.gadget.wiwosm","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","oojs-ui.styles.icons-media","oojs-ui-core.icons","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=hu&modules=ext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=hu&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=hu&modules=ext.gadget.infobox%2CwikiMenuStyles&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=hu&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Hilbert-tér – Wikipédia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//hu.m.wikipedia.org/wiki/Hilbert-t%C3%A9r"> <link rel="alternate" type="application/x-wiki" title="Szerkesztés" href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipédia (hu)"> <link rel="EditURI" type="application/rsd+xml" href="//hu.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://hu.wikipedia.org/wiki/Hilbert-t%C3%A9r"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.hu"> <link rel="alternate" type="application/atom+xml" title="Wikipédia Atom-hírcsatorna" href="/w/index.php?title=Speci%C3%A1lis:Friss_v%C3%A1ltoztat%C3%A1sok&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Hilbert-tér rootpage-Hilbert-tér skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Ugrás a tartalomhoz</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Wiki"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Főmenü" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Főmenü</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Főmenü</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">elrejtés</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigáció </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Kezd%C5%91lap" title="A kezdőlap megtekintése [z]" accesskey="z"><span>Kezdőlap</span></a></li><li id="n-sidebar-contents" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Tartalom"><span>Tartalom</span></a></li><li id="n-sidebar-featured" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Kiemelt_sz%C3%B3cikkek_list%C3%A1ja"><span>Kiemelt szócikkek</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Friss_v%C3%A1ltoztat%C3%A1sok" title="A wikiben történt legutóbbi változtatások listája [r]" accesskey="r"><span>Friss változtatások</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Lap_tal%C3%A1lomra" title="Egy véletlenszerűen kiválasztott lap betöltése [x]" accesskey="x"><span>Lap találomra</span></a></li><li id="n-sidebar-enquiries" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Tudakoz%C3%B3"><span>Tudakozó</span></a></li> </ul> </div> </div> <div id="p-sidebar-participate" class="vector-menu mw-portlet mw-portlet-sidebar-participate" > <div class="vector-menu-heading"> Részvétel </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-sidebar-basics" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:%C3%9Aj_szerkeszt%C5%91knek"><span>Kezdőknek</span></a></li><li id="n-sidebar-help" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Seg%C3%ADts%C3%A9g"><span>Segítség</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Port%C3%A1l:K%C3%B6z%C3%B6ss%C3%A9g" title="A projektről, miben segíthetsz, mit hol találsz meg"><span>Közösségi portál</span></a></li><li id="n-sidebar-contact" class="mw-list-item"><a href="/wiki/Wikip%C3%A9dia:Kapcsolatfelv%C3%A9tel"><span>Kapcsolatfelvétel</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Kezd%C5%91lap" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipédia" src="/static/images/mobile/copyright/wikipedia-wordmark-fr.svg" style="width: 7.4375em; height: 1.125em;"> <img class="mw-logo-tagline" alt="" src="/static/images/mobile/copyright/wikipedia-tagline-hu.svg" width="120" height="13" style="width: 7.5em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Speci%C3%A1lis:Keres%C3%A9s" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Keresés a Wikipédián [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Keresés</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Keresés a Wikipédián" aria-label="Keresés a Wikipédián" autocapitalize="sentences" title="Keresés a Wikipédián [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Speciális:Keresés"> </div> <button class="cdx-button cdx-search-input__end-button">Keresés</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Személyes eszközök"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Megjelenés"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Megjelenés" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Megjelenés</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hu.wikipedia.org&uselang=hu" class=""><span>Adományok</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speci%C3%A1lis:Szerkeszt%C5%91i_fi%C3%B3k_l%C3%A9trehoz%C3%A1sa&returnto=Hilbert-t%C3%A9r" title="Arra bíztatunk, hogy hozz létre egy fiókot, és jelentkezz be, azonban ez nem kötelező" class=""><span>Fiók létrehozása</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Speci%C3%A1lis:Bel%C3%A9p%C3%A9s&returnto=Hilbert-t%C3%A9r" title="Bejelentkezni javasolt, de nem kötelező [o]" accesskey="o" class=""><span>Bejelentkezés</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="További lehetőségek" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Személyes eszközök" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Személyes eszközök</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Felhasználói menü" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_hu.wikipedia.org&uselang=hu"><span>Adományok</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Szerkeszt%C5%91i_fi%C3%B3k_l%C3%A9trehoz%C3%A1sa&returnto=Hilbert-t%C3%A9r" title="Arra bíztatunk, hogy hozz létre egy fiókot, és jelentkezz be, azonban ez nem kötelező"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Fiók létrehozása</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:Bel%C3%A9p%C3%A9s&returnto=Hilbert-t%C3%A9r" title="Bejelentkezni javasolt, de nem kötelező [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Bejelentkezés</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Lapok kijelentkezett szerkesztőknek <a href="/wiki/Seg%C3%ADts%C3%A9g:Bevezet%C3%A9s" aria-label="Tudj meg többet a szerkesztésről"><span>további információk</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:K%C3%B6zrem%C5%B1k%C3%B6d%C3%A9seim" title="Erről az IP-címről végrehajtott szerkesztések listája [y]" accesskey="y"><span>Közreműködések</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Vit%C3%A1m" title="Az általad használt IP-címről végrehajtott szerkesztések megvitatása [n]" accesskey="n"><span>Vitalap</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Wiki"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Tartalomjegyzék" 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-Bevezetés" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bevezetés"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Bevezetés</span> </div> </a> <ul id="toc-Bevezetés-sublist" class="vector-toc-list"> </ul> </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">2</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">3</span> <span>Példák</span> </div> </a> <ul id="toc-Példák-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ortogonalitás" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ortogonalitás"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ortogonalitás</span> </div> </a> <ul id="toc-Ortogonalitás-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bázis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bázis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bázis</span> </div> </a> <ul id="toc-Bázis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alterek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alterek"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Alterek</span> </div> </a> <ul id="toc-Alterek-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konjugált_Hilbert-tér" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konjugált_Hilbert-tér"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Konjugált Hilbert-tér</span> </div> </a> <ul id="toc-Konjugált_Hilbert-tér-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert-terek_közötti_leképezések" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hilbert-terek_közötti_leképezések"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Hilbert-terek közötti leképezések</span> </div> </a> <ul id="toc-Hilbert-terek_közötti_leképezések-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riesz_reprezentációs_tétel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Riesz_reprezentációs_tétel"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Riesz reprezentációs tétel</span> </div> </a> <ul id="toc-Riesz_reprezentációs_tétel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier-együttható" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fourier-együttható"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Fourier-együttható</span> </div> </a> <ul id="toc-Fourier-együttható-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-RKHS" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#RKHS"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>RKHS</span> </div> </a> <ul id="toc-RKHS-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alkalmazások" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alkalmazások"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Alkalmazások</span> </div> </a> <ul id="toc-Alkalmazások-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Érdekesség" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Érdekesség"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Érdekesség</span> </div> </a> <ul id="toc-Érdekesség-sublist" class="vector-toc-list"> </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">14</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-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">15</span> <span>Fordítás</span> </div> </a> <ul id="toc-Fordítás-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">16</span> <span>Források</span> </div> </a> <ul id="toc-Források-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-További_információk" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#További_információk"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>További információk</span> </div> </a> <ul id="toc-További_információk-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">Hilbert-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 59 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-59" 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">59 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 badge-Q17437798 badge-goodarticle mw-list-item" title="jó szócikk"><a href="https://en.wikipedia.org/wiki/Hilbert_space" title="Hilbert space – angol" lang="en" hreflang="en" data-title="Hilbert space" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hilbert-ruimte" title="Hilbert-ruimte – afrikaans" lang="af" hreflang="af" data-title="Hilbert-ruimte" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</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%87%D9%8A%D9%84%D8%A8%D8%B1%D8%AA" title="فضاء هيلبرت – arab" lang="ar" hreflang="ar" data-title="فضاء هيلبرت" data-language-autonym="العربية" data-language-local-name="arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_de_Hilbert" title="Espaciu de Hilbert – asztúr" lang="ast" hreflang="ast" data-title="Espaciu de Hilbert" data-language-autonym="Asturianu" data-language-local-name="asztúr" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Hilbert_f%C9%99zas%C4%B1" title="Hilbert fəzası – azerbajdzsáni" lang="az" hreflang="az" data-title="Hilbert fəzası" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzsáni" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%93%D1%8B" title="Гильберт арауығы – baskír" lang="ba" hreflang="ba" data-title="Гильберт арауығы" data-language-autonym="Башҡортса" data-language-local-name="baskír" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BB%D0%B1%D0%B5%D1%80%D1%82%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Хилбертово пространство – bolgár" lang="bg" hreflang="bg" data-title="Хилбертово пространство" data-language-autonym="Български" data-language-local-name="bolgár" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A6%BF%E0%A6%B2%E0%A6%AC%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%9F_%E0%A6%9C%E0%A6%97%E0%A7%8E" title="হিলবার্ট জগৎ – bangla" lang="bn" hreflang="bn" data-title="হিলবার্ট জগৎ" data-language-autonym="বাংলা" data-language-local-name="bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_de_Hilbert" title="Espai de Hilbert – katalán" lang="ca" hreflang="ca" data-title="Espai de Hilbert" data-language-autonym="Català" data-language-local-name="katalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%DA%BE%DB%8C%D9%84%D8%A8%DB%8E%D8%B1%D8%AA" title="بۆشاییی ھیلبێرت – közép-ázsiai kurd" lang="ckb" hreflang="ckb" data-title="بۆشاییی ھیلبێرت" data-language-autonym="کوردی" data-language-local-name="közép-ázsiai kurd" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hilbert%C5%AFv_prostor" title="Hilbertův prostor – cseh" lang="cs" hreflang="cs" data-title="Hilbertův prostor" data-language-autonym="Čeština" data-language-local-name="cseh" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hilbertrum" title="Hilbertrum – dán" lang="da" hreflang="da" data-title="Hilbertrum" 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/Hilbertraum" title="Hilbertraum – német" lang="de" hreflang="de" data-title="Hilbertraum" data-language-autonym="Deutsch" data-language-local-name="német" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A7%CF%8E%CF%81%CE%BF%CF%82_%CE%A7%CE%AF%CE%BB%CE%BC%CF%80%CE%B5%CF%81%CF%84" title="Χώρος Χίλμπερτ – görög" lang="el" hreflang="el" data-title="Χώρος Χίλμπερτ" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Hilberta_spaco" title="Hilberta spaco – eszperantó" lang="eo" hreflang="eo" data-title="Hilberta spaco" data-language-autonym="Esperanto" data-language-local-name="eszperantó" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_de_Hilbert" title="Espacio de Hilbert – spanyol" lang="es" hreflang="es" data-title="Espacio de Hilbert" 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/Hilberti_ruum" title="Hilberti ruum – észt" lang="et" hreflang="et" data-title="Hilberti ruum" data-language-autonym="Eesti" data-language-local-name="észt" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Hilberten_espazio" title="Hilberten espazio – baszk" lang="eu" hreflang="eu" data-title="Hilberten espazio" data-language-autonym="Euskara" data-language-local-name="baszk" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%87%DB%8C%D9%84%D8%A8%D8%B1%D8%AA" title="فضای هیلبرت – perzsa" lang="fa" hreflang="fa" data-title="فضای هیلبرت" 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/Hilbertin_avaruus" title="Hilbertin avaruus – finn" lang="fi" hreflang="fi" data-title="Hilbertin avaruus" 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_de_Hilbert" title="Espace de Hilbert – francia" lang="fr" hreflang="fr" data-title="Espace de Hilbert" data-language-autonym="Français" data-language-local-name="francia" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_de_Hilbert" title="Espazo de Hilbert – gallego" lang="gl" hreflang="gl" data-title="Espazo de Hilbert" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</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%94%D7%99%D7%9C%D7%91%D7%A8%D7%98" title="מרחב הילברט – héber" lang="he" hreflang="he" data-title="מרחב הילברט" data-language-autonym="עברית" data-language-local-name="héber" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%AC%D5%A2%D5%A5%D6%80%D5%BF%D5%B5%D5%A1%D5%B6_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հիլբերտյան տարածություն – örmény" lang="hy" hreflang="hy" data-title="Հիլբերտյան տարածություն" data-language-autonym="Հայերեն" data-language-local-name="örmény" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_Hilbert" title="Ruang Hilbert – indonéz" lang="id" hreflang="id" data-title="Ruang Hilbert" 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/Hilbert-r%C3%BAm" title="Hilbert-rúm – izlandi" lang="is" hreflang="is" data-title="Hilbert-rúm" 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_di_Hilbert" title="Spazio di Hilbert – olasz" lang="it" hreflang="it" data-title="Spazio di Hilbert" 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%92%E3%83%AB%E3%83%99%E3%83%AB%E3%83%88%E7%A9%BA%E9%96%93" title="ヒルベルト空間 – japán" lang="ja" hreflang="ja" data-title="ヒルベルト空間" 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/%ED%9E%90%EB%B2%A0%EB%A5%B4%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="힐베르트 공간 – koreai" lang="ko" hreflang="ko" data-title="힐베르트 공간" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82_%D0%BC%D0%B5%D0%B9%D0%BA%D0%B8%D0%BD%D0%B4%D0%B8%D0%B3%D0%B8" title="Гильберт мейкиндиги – kirgiz" lang="ky" hreflang="ky" data-title="Гильберт мейкиндиги" data-language-autonym="Кыргызча" data-language-local-name="kirgiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Hilberto_erdv%C4%97" title="Hilberto erdvė – litván" lang="lt" hreflang="lt" data-title="Hilberto erdvė" data-language-autonym="Lietuvių" data-language-local-name="litván" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BB%D0%B1%D0%B5%D1%80%D1%82%D0%B8%D0%B9%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD_%D0%B7%D0%B0%D0%B9" title="Хилбертийн орон зай – mongol" lang="mn" hreflang="mn" data-title="Хилбертийн орон зай" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_Hilbert" title="Ruang Hilbert – maláj" lang="ms" hreflang="ms" data-title="Ruang Hilbert" data-language-autonym="Bahasa Melayu" data-language-local-name="maláj" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hilbertruimte" title="Hilbertruimte – holland" lang="nl" hreflang="nl" data-title="Hilbertruimte" data-language-autonym="Nederlands" data-language-local-name="holland" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hilbertrom" title="Hilbertrom – norvég (nynorsk)" lang="nn" hreflang="nn" data-title="Hilbertrom" data-language-autonym="Norsk nynorsk" data-language-local-name="norvég (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hilbert-rom" title="Hilbert-rom – norvég (bokmål)" lang="nb" hreflang="nb" data-title="Hilbert-rom" data-language-autonym="Norsk bokmål" data-language-local-name="norvég (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B9%E0%A8%BF%E0%A8%B2%E0%A8%AC%E0%A8%B0%E0%A8%9F_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਹਿਲਬਰਟ ਸਪੇਸ – pandzsábi" lang="pa" hreflang="pa" data-title="ਹਿਲਬਰਟ 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_Hilberta" title="Przestrzeń Hilberta – lengyel" lang="pl" hreflang="pl" data-title="Przestrzeń Hilberta" data-language-autonym="Polski" data-language-local-name="lengyel" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DB%81%D9%84%D8%A8%D8%B1%D9%B9_%D8%B3%D9%BE%DB%8C%D8%B3" title="ہلبرٹ سپیس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ہلبرٹ سپیس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_de_Hilbert" title="Espaço de Hilbert – portugál" lang="pt" hreflang="pt" data-title="Espaço de Hilbert" 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_Hilbert" title="Spațiu Hilbert – román" lang="ro" hreflang="ro" data-title="Spațiu Hilbert" 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%93%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Гильбертово пространство – orosz" lang="ru" hreflang="ru" data-title="Гильбертово пространство" data-language-autonym="Русский" data-language-local-name="orosz" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Hilbert_space" title="Hilbert space – skót" lang="sco" hreflang="sco" data-title="Hilbert space" data-language-autonym="Scots" data-language-local-name="skót" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Hilbertov_prostor" title="Hilbertov prostor – szerbhorvát" lang="sh" hreflang="sh" data-title="Hilbertov prostor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="szerbhorvát" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hilbert_space" title="Hilbert space – Simple English" lang="en-simple" hreflang="en-simple" data-title="Hilbert space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hilbertov_priestor" title="Hilbertov priestor – szlovák" lang="sk" hreflang="sk" data-title="Hilbertov priestor" data-language-autonym="Slovenčina" data-language-local-name="szlovák" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hilbertov_prostor" title="Hilbertov prostor – szlovén" lang="sl" hreflang="sl" data-title="Hilbertov prostor" data-language-autonym="Slovenščina" data-language-local-name="szlovén" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hap%C3%ABsira_e_Hilbertit" title="Hapësira e Hilbertit – albán" lang="sq" hreflang="sq" data-title="Hapësira e Hilbertit" data-language-autonym="Shqip" data-language-local-name="albán" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BB%D0%B1%D0%B5%D1%80%D1%82%D0%BE%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Хилбертов простор – szerb" lang="sr" hreflang="sr" data-title="Хилбертов простор" data-language-autonym="Српски / srpski" data-language-local-name="szerb" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hilbertrum" title="Hilbertrum – svéd" lang="sv" hreflang="sv" data-title="Hilbertrum" data-language-autonym="Svenska" data-language-local-name="svéd" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espasyong_Hilbert" title="Espasyong Hilbert – tagalog" lang="tl" hreflang="tl" data-title="Espasyong Hilbert" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hilbert_uzay%C4%B1" title="Hilbert uzayı – török" lang="tr" hreflang="tr" data-title="Hilbert 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%93%D1%96%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82%D1%96%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Гільбертів простір – ukrán" lang="uk" hreflang="uk" data-title="Гільбертів простір" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gilbert_fazosi" title="Gilbert fazosi – üzbég" lang="uz" hreflang="uz" data-title="Gilbert fazosi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="üzbég" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_Hilbert" title="Không gian Hilbert – vietnámi" lang="vi" hreflang="vi" data-title="Không gian Hilbert" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4" title="希尔伯特空间 – wu kínai" lang="wuu" hreflang="wuu" data-title="希尔伯特空间" data-language-autonym="吴语" data-language-local-name="wu kínai" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4" title="希尔伯特空间 – kínai" lang="zh" hreflang="zh" data-title="希尔伯特空间" 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/%E5%B8%8C%E7%88%BE%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%96%93" title="希爾伯特空間 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="希爾伯特空間" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9B%82%E6%8B%94%E7%A9%BA%E9%96%93" title="囂拔空間 – kantoni" lang="yue" hreflang="yue" data-title="囂拔空間" data-language-autonym="粵語" data-language-local-name="kantoni" class="interlanguage-link-target"><span>粵語</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/Q190056#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/Hilbert-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:Hilbert-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/Hilbert-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=Hilbert-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=Hilbert-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/Hilbert-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=Hilbert-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=Hilbert-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/Hilbert-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/Hilbert-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=Hilbert-t%C3%A9r&oldid=27407869" 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=Hilbert-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=Hilbert-t%C3%A9r&id=27407869&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%2FHilbert-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%2FHilbert-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=Hilbert-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=Hilbert-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=Hilbert-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:Hilbert_space" 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/Q190056" 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=Hilbert-t%C3%A9r">ellenőrizve</a>: <i>2024. augusztus 30.</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"><table class="dablink noprint noviewer" style="padding-left: 2em; vertical-align: middle;" cellpadding="0" cellspacing="0"><tbody><tr><td style="padding-right:.25em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Disambig.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/19px-Disambig.svg.png" decoding="async" width="19" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/29px-Disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/38px-Disambig.svg.png 2x" data-file-width="230" data-file-height="183" /></a></span></td><td><i>A „Tér” lehetséges további jelentéseiről lásd: <a href="/wiki/T%C3%A9r_(egy%C3%A9rtelm%C5%B1s%C3%ADt%C5%91_lap)" class="mw-disambig" title="Tér (egyértelműsítő lap)">Tér (egyértelműsítő lap)</a>.</i></td></tr></tbody></table> <p>A <b>Hilbert-tér</b> a modern <a href="/wiki/Matematika" title="Matematika">matematika</a> fontos fogalma: olyan <a href="/wiki/Skal%C3%A1rszorzatos_vektort%C3%A9r" title="Skalárszorzatos vektortér">skalárszorzatos vektortér</a>, amely <a href="/w/index.php?title=Teljes_t%C3%A9r&action=edit&redlink=1" class="new" title="Teljes tér (a lap nem létezik)">teljes</a> a <a href="/wiki/Skal%C3%A1rszorzat" class="mw-redirect" title="Skalárszorzat">skalárszorzat</a> által definiált <a href="/wiki/Norm%C3%A1lt_t%C3%A9r" title="Normált tér">normára</a> nézve. A Hilbert-tereket a <a href="/wiki/Funkcion%C3%A1lanal%C3%ADzis" title="Funkcionálanalízis">funkcionálanalízis</a> tanulmányozza. A Hilbert-térnek alapvető jelentősége van a <a href="/wiki/Kvantummechanika" title="Kvantummechanika">kvantummechanika</a> megalapozásában, jóllehet a kvantummechanika sok alapvető tulajdonsága megérthető a Hilbert-terek mélyebb megértése nélkül.<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> A Hilbert-tér egyben <a href="/wiki/Banach-t%C3%A9r" title="Banach-tér">Banach-tér</a> is, melynek normáját skalárszorzat <a href="/w/index.php?title=Induk%C3%A1lt_norma&action=edit&redlink=1" class="new" title="Indukált norma (a lap nem létezik)">indukálja</a>. </p><p>Szerkezetét egyértelműen meghatározza a Hilbert-dimenziója. Ez tetszőleges <a href="/wiki/Kardin%C3%A1lis_sz%C3%A1m" title="Kardinális szám">kardinális szám</a> lehet. Ha a dimenzió véges, akkor euklideszi vektortérről van szó. Sok területen, például a kvantummechanikában a megszámlálhatóan végtelen dimenziós Hilbert-teret használják. A Hilbert-tér egy eleme megadható a dimenziónak megfelelő számú valós, vagy komplex koordinátával. A vektorterekhez hasonlóan, ahol egy Hamel-bázisban megadott koordináták véges kivétellel nullák, egy Hilbert-tér ortonormált bázisában csak megszámlálható sok koordináta különbözhet nullától, és a koordináták négyzetesen összegezhetők. </p><p>A Hilbert-tereken értelmezett skalárszorzat topologikus szerkezettel is ellátja a teret; ez lehetővé teszi a határértékek megközelítését, szemben az általános vektorterekkel. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Bevezetés"><span id="Bevezet.C3.A9s"></span>Bevezetés</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=1" title="Szakasz szerkesztése: Bevezetés"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/F%C3%A1jl:Hilbert.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Hilbert.jpg/160px-Hilbert.jpg" decoding="async" width="160" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Hilbert.jpg/240px-Hilbert.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Hilbert.jpg/320px-Hilbert.jpg 2x" data-file-width="437" data-file-height="592" /></a><figcaption>David Hilbert</figcaption></figure> <p>A Hilbert-teret <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbertről</a> nevezték el, aki az integrálegyenletekkel kapcsolatban tanulmányozta azokat. Az elnevezés eredete „der abstrakte Hilbertsche Raum” <a href="/wiki/Neumann_J%C3%A1nos" title="Neumann János">Neumann Jánostól</a> származik, a nemkorlátos hermitikus operátorokról szóló 1929-es híres cikkéből. Neumann volt talán az a matematikus, aki legtisztábban látta a jelentőségét, annak a megtermékenyítően ható munkájának következtében, mellyel a kvantummechanikát szilárd alapokra helyezte. A „Hilbert-tér” elnevezést hamarosan mások is elfogadták, például <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> az 1931-ben publikált <i>A csoportok és a kvantummechanika elmélete</i> <i>(The Theory of Groups and Quantum Mechanics)</i> című könyvében. </p><p>Az absztrakt Hilbert-tér elemeit „vektoroknak” nevezik. A kvantummechanikában például egy fizikai rendszert egy „<a href="/wiki/Hull%C3%A1mf%C3%BCggv%C3%A9ny" title="Hullámfüggvény">hullámfüggvényekből</a>” álló komplex Hilbert-tér ír le, mely hullámfüggvények a rendszer egyes állapotait írják le, a hullámfüggvények egy <a href="/w/index.php?title=L-p_t%C3%A9r&action=edit&redlink=1" class="new" title="L-p tér (a lap nem létezik)">L-2-tér</a> elemei a kvantummechanika modern megfogalmazásában. A kvantummechanikában gyakran használt síkhullámok és kötött állapotok Hilbert-terére a formálisabb <a href="/w/index.php?title=Kifesz%C3%ADtett_Hilbert-t%C3%A9r&action=edit&redlink=1" class="new" title="Kifeszített Hilbert-tér (a lap nem létezik)">kifeszített Hilbert-tér</a> néven hivatkoznak. </p> <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=Hilbert-t%C3%A9r&action=edit&section=2" title="Szakasz szerkesztése: Definíció"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>H</i> <a href="/wiki/Vektort%C3%A9r" title="Vektortér">vektorteret</a> a <i>T</i> test (valós vagy komplex számtest) feletti <b>Hilbert-tér</b>nek nevezzük, ha értelmezve van rajta egy <i>Hermite-féle alak</i> (belső szorzat), amely egy teljes <a href="/wiki/Norm%C3%A1lt_t%C3%A9r" title="Normált tér">normált teret</a> indukál. </p><p>Azaz létezik egy leképzés: <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 \langle \cdot ,\cdot \rangle \colon H\times H\to T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>:</mo> <mi>H</mi> <mo>×</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle \colon H\times H\to T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3e03312b6e6a9927a4e46891d022c7c1e32673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.389ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle \colon H\times H\to T}"></span>, amely 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>-beli <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>, <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>, <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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-re és 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>-beli <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>-ra a következőket teljesíti: </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 \langle {x},{x}\rangle \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {x},{x}\rangle \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e516c2a42d2c021b61c16e62b4e37835143ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle \langle {x},{x}\rangle \geq 0}"></span> (nemnegatív);</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 \langle {x},{x}\rangle =0\Leftrightarrow {x}={0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {x},{x}\rangle =0\Leftrightarrow {x}={0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68121ff63cee2726972f4c6022f54af9d28432e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.968ex; height:2.843ex;" alt="{\displaystyle \langle {x},{x}\rangle =0\Leftrightarrow {x}={0}}"></span> (definit);</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 \langle {x},{y}\rangle ={\overline {\langle {y},{x}\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {x},{y}\rangle ={\overline {\langle {y},{x}\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1e3ecf20f4a19102506ef89c077ff5836f95bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.87ex; height:3.676ex;" alt="{\displaystyle \langle {x},{y}\rangle ={\overline {\langle {y},{x}\rangle }}}"></span> (hermitikus - valós esetben a konjugálás elhagyható);</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 \langle {x},\lambda {y}\rangle =\lambda \langle {x},{y}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>λ</mi> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {x},\lambda {y}\rangle =\lambda \langle {x},{y}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b9a4991e6639d94c7331ad7b3fe5487a957f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.466ex; height:2.843ex;" alt="{\displaystyle \langle {x},\lambda {y}\rangle =\lambda \langle {x},{y}\rangle }"></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 \langle {x},{y}+{z}\rangle =\langle {x},{y}\rangle +\langle {x},{z}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {x},{y}+{z}\rangle =\langle {x},{y}\rangle +\langle {x},{z}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/557df92b654d1bee6822f59b97b401e10ceea6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.785ex; height:2.843ex;" alt="{\displaystyle \langle {x},{y}+{z}\rangle =\langle {x},{y}\rangle +\langle {x},{z}\rangle }"></span> (lineáris a <i>második</i> argumentumban).</li></ol> <p>Minden, az előbbi tulajdonságokat teljesítő, belső szorzatos térben értelmezhető egy ||.|| <a href="/wiki/Norm%C3%A1lt_t%C3%A9r" title="Normált tér">norma</a> következőképpen: </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><i>H</i> Hilbert-tér, ha <i>H</i> erre a normára nézve teljes, azaz minden <i>H</i>-beli <a href="/wiki/Cauchy-sorozat" title="Cauchy-sorozat">Cauchy-sorozat</a> <a href="/wiki/Konvergencia_(matematika)" title="Konvergencia (matematika)">konvergál</a>. </p><p><i>Megjegyzések:</i> </p> <ul><li>Ebben a definícióban a skaláris szorzat a második argumentumban lineáris, az elsőben C-<a href="/w/index.php?title=Antiline%C3%A1ris_lek%C3%A9pez%C3%A9s&action=edit&redlink=1" class="new" title="Antilineáris leképezés (a lap nem létezik)">antilineáris</a>, ez fordítva is működne, illetve használják is.</li> <li>Ha egy Hermite-féle alak értelmezve van a komplex vektortéren, akkor <a href="/w/index.php?title=Unit%C3%A9r_t%C3%A9r&action=edit&redlink=1" class="new" title="Unitér tér (a lap nem létezik)">unitér térről</a> beszélünk, valós esetben <a href="/wiki/Euklideszi_t%C3%A9r_(line%C3%A1ris_algebra)" title="Euklideszi tér (lineáris algebra)">euklideszi vektortérről</a>.</li> <li>Mivel minden Hilbert-tér unitér vagy euklideszi, érvényes benne a <a href="/wiki/Cauchy%E2%80%93Bunyakovszkij%E2%80%93Schwarz-egyenl%C5%91tlens%C3%A9g" title="Cauchy–Bunyakovszkij–Schwarz-egyenlőtlenség">Cauchy–Bunyakovszkij–Schwarz-egyenlőtlenség</a>, a <a href="/wiki/Polariz%C3%A1ci%C3%B3s_formula" title="Polarizációs formula">polarizációs formula</a> és a <a href="/wiki/Paralelogrammaazonoss%C3%A1g" title="Paralelogrammaazonosság">paralelogrammaazonosság</a>, valamint a <a href="/wiki/Bessel-egyenl%C5%91tlens%C3%A9g" title="Bessel-egyenlőtlenség">Bessel-egyenlőtlenség</a> és a <a href="/wiki/Pitagorasz-t%C3%A9tel" title="Pitagorasz-tétel">Pitagorasz-tétel</a>.</li> <li>A 3. tulajdonságban a felülvonás a komplex konjugálást jelöli, valós esetben a 3. tulajdonság a skaláris szorzat szimmetriája.</li> <li>Vegyük észre, hogy a norma definíciójában a gyök a norma homogenitása miatt szükséges.</li></ul> <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=Hilbert-t%C3%A9r&action=edit&section=3" title="Szakasz szerkesztése: Példák"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> koordinátatér 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 \langle u,v\rangle =u_{1}v_{1}+\dotsb +u_{n}v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle =u_{1}v_{1}+\dotsb +u_{n}v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc09a8bf2430e8e6ec7dbf63b64515398805db91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.263ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle =u_{1}v_{1}+\dotsb +u_{n}v_{n}}"></span> valós skalárszorzattal</li> <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 \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> koordinátatér 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 \langle u,v\rangle ={\bar {u}}_{1}v_{1}+\dotsb +{\bar {u}}_{n}v_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle ={\bar {u}}_{1}v_{1}+\dotsb +{\bar {u}}_{n}v_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807caafd2c8699390833d439a6caf561fff55102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.263ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle ={\bar {u}}_{1}v_{1}+\dotsb +{\bar {u}}_{n}v_{n}}"></span> skalárszorzattal</li> <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 {\mathbb {K} }^{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {K} }^{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0dbacca0686fd166dcc3edc7a3cd7668929802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.748ex; height:2.343ex;" alt="{\displaystyle {\mathbb {K} }^{m\times n}}"></span> valós vagy komplex mátrixtér a <a href="/w/index.php?title=Frobenius-skal%C3%A1rszorzat&action=edit&redlink=1" class="new" title="Frobenius-skalárszorzat (a lap nem létezik)">Frobenius-skalárszorzattal</a></li> <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 H^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1e5feedc666fbe45841f42f671a84565144f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.163ex; height:2.343ex;" alt="{\displaystyle H^{p}}"></span> Szoboljev-tér 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 p\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dccf79a2af14bcbd9b24fd7d719e48849a22f713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\geq 0}"></span> esetén. Ezek képezik a parciális differenciálegyenletek megoldáselméletének alapját.</li> <li>A <a href="/w/index.php?title=Hilbert-Schmidt-oper%C3%A1tor&action=edit&redlink=1" class="new" title="Hilbert-Schmidt-operátor (a lap nem létezik)">Hilbert-Schmidt-operátorok</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 HS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle HS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0161fa27be2404ec38687965c474cb74756bf6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.176ex;" alt="{\displaystyle HS}"></span> tere.</li> <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 H^{2}(\mathbb {D} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{2}(\mathbb {D} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737e02b4b2bca29a06835fbe8f2f2ab11366e490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:3.176ex;" alt="{\displaystyle H^{2}(\mathbb {D} )}"></span> <a href="/w/index.php?title=Hardy-t%C3%A9r&action=edit&redlink=1" class="new" title="Hardy-tér (a lap nem létezik)">Hardy-tér</a> és 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 {\mathcal {H}}^{2}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}^{2}(\mathbb {R} ^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d755bc8216cccb076880a3e9af12952b596e8eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.724ex; height:3.176ex;" alt="{\displaystyle {\mathcal {H}}^{2}(\mathbb {R} ^{n})}"></span> <a href="/w/index.php?title=Val%C3%B3s_Hardy-t%C3%A9r&action=edit&redlink=1" class="new" title="Valós Hardy-tér (a lap nem létezik)">valós Hardy-tér</a>.</li> <li>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 \ell ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}"></span> <a href="/w/index.php?title=Sorozatt%C3%A9r&action=edit&redlink=1" class="new" title="Sorozattér (a lap nem létezik)">sorozattér</a>, melyet azok a sorozatok alkotnak, ahol a sorozat elemeinek négyzetösszege véges. David Hilbert ezt a teret vizsgálta. Fontossága abban áll, hogy minden <a href="/w/index.php?title=Szepar%C3%A1bilis_t%C3%A9r&action=edit&redlink=1" class="new" title="Szeparábilis tér (a lap nem létezik)">szeparábilis</a> végtelen dimenziós Hilbert-tér <a href="/w/index.php?title=Izometrikus_izomorfia&action=edit&redlink=1" class="new" title="Izometrikus izomorfia (a lap nem létezik)">izometrikusan izomorf</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 \ell ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}"></span>-tel.</li> <li>A <a href="/w/index.php?title=N%C3%A9gyzetesen_integr%C3%A1lhat%C3%B3_f%C3%BCggv%C3%A9nyek_tere&action=edit&redlink=1" class="new" title="Négyzetesen integrálható függvények tere (a lap nem létezik)">négyzetesen integrálható függvények</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 L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"></span> 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 \textstyle \langle f,g\rangle _{L^{2}}=\int {\overline {f(x)}}\,g(x)\,{\rm {d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \langle f,g\rangle _{L^{2}}=\int {\overline {f(x)}}\,g(x)\,{\rm {d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24860926ccced250629c935b3a03bd490cf74e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.508ex; height:3.843ex;" alt="{\displaystyle \textstyle \langle f,g\rangle _{L^{2}}=\int {\overline {f(x)}}\,g(x)\,{\rm {d}}x}"></span> skalárszorzattal.</li> <li>A <a href="/w/index.php?title=Majdnemperiodikus_f%C3%BCggv%C3%A9ny&action=edit&redlink=1" class="new" title="Majdnemperiodikus függvény (a lap nem létezik)">majdnemperiodikus függvények</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 \mathrm {AP} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AP} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7deea8958a54c26cf69de67f8433708b39934bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.38ex; height:2.676ex;" alt="{\displaystyle \mathrm {AP} ^{2}}"></span> tere, ami a következőképpen definiálható: 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 \lambda \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87bc1622689bc998795834cd65eecdb4955a785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {R} }"></span>, ehhez tekintjük azokat 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 f_{\lambda }\colon \mathbb {R} \to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\lambda }\colon \mathbb {R} \to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c652737c27f45ef45f9a5e8e5dad4a9e69deae3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.334ex; height:2.509ex;" alt="{\displaystyle f_{\lambda }\colon \mathbb {R} \to \mathbb {C} }"></span> függvényeket, 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 f_{\lambda }\left(t\right)=e^{i\lambda t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>λ</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\lambda }\left(t\right)=e^{i\lambda t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46bc4ce91f62975266e99b5510eb21f5de83b5d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.899ex; height:3.176ex;" alt="{\displaystyle f_{\lambda }\left(t\right)=e^{i\lambda t}}"></span>. Ellátjuk 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 {lin} \left\{f_{\lambda }\colon \lambda \in \mathbb {R} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>lin</mi> <mo>⁡</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>:</mo> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {lin} \left\{f_{\lambda }\colon \lambda \in \mathbb {R} \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c621bfb8e575db4bdc7a66fa372394ef7f60327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.149ex; height:2.843ex;" alt="{\displaystyle \operatorname {lin} \left\{f_{\lambda }\colon \lambda \in \mathbb {R} \right\}}"></span> teret 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 \textstyle \langle f,g\rangle =\lim _{T\to +\infty }{\tfrac {1}{4T}}\int _{-T}^{T}{\overline {f(t)}}\,g(t)\,{\rm {d}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>T</mi> </mrow> </mfrac> </mstyle> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \langle f,g\rangle =\lim _{T\to +\infty }{\tfrac {1}{4T}}\int _{-T}^{T}{\overline {f(t)}}\,g(t)\,{\rm {d}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c912ee5e5870fc9013c109ce41c5bb92fffd06b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.975ex; height:4.009ex;" alt="{\displaystyle \textstyle \langle f,g\rangle =\lim _{T\to +\infty }{\tfrac {1}{4T}}\int _{-T}^{T}{\overline {f(t)}}\,g(t)\,{\rm {d}}t}"></span> skalárszorzattal, így prehilbertteret kapunk. Ezt a teret teljessé téve jutunk 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 \mathrm {AP} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {AP} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7deea8958a54c26cf69de67f8433708b39934bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.38ex; height:2.676ex;" alt="{\displaystyle \mathrm {AP} ^{2}}"></span> Hilbert-térhez, ami nem szeparábilis.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ortogonalitás"><span id="Ortogonalit.C3.A1s"></span>Ortogonalitás</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=4" title="Szakasz szerkesztése: Ortogonalitás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Két vektort <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\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/947e70f01d5562fb467fb8f05837045fa126108a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.423ex; height:2.509ex;" alt="{\displaystyle x,y\in H}"></span> ortogonálisnak mondunk, 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 \langle {x},{y}\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {x},{y}\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10193b040a6495a3fce1a0380aacd684e0c43842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.589ex; height:2.843ex;" alt="{\displaystyle \langle {x},{y}\rangle =0}"></span>, gyakori jelölés: <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\perp y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊥</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\perp y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36702069481de67a3e4659e380c6ac7c67d0f4ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.509ex;" alt="{\displaystyle x\perp y}"></span>. </p><p>Egy <i>S</i> halmazt <i>H</i>-beli ortogonális rendszernek nevezünk, 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 S\subset H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b29f290397b3868fcdb875df4cbc7840a5314a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.661ex; height:2.176ex;" alt="{\displaystyle S\subset H}"></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 \forall x,y\in S,x\neq y:\langle {x},{y}\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in S,x\neq y:\langle {x},{y}\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4581a551bb07830fdeea19507826fad2210794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.296ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in S,x\neq y:\langle {x},{y}\rangle =0}"></span>. Ha egy ortogonális rendszer nem bővíthető (maximális), akkor ortogonális bázis. Az ortogonális bázisok lineáris burka sűrű a Hilbert-térben. A lineáris algebrában megszokott értelemben ezek csak véges dimenziós esetben bázisok. </p><p>Egy <i>S</i> halmazt <i>H</i>-beli ortonormált rendszernek nevezünk, 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 S\subset H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b29f290397b3868fcdb875df4cbc7840a5314a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.661ex; height:2.176ex;" alt="{\displaystyle S\subset H}"></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 \forall x_{i},x_{j}\in S:\langle {x_{i}},{x_{j}}\rangle =\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x_{i},x_{j}\in S:\langle {x_{i}},{x_{j}}\rangle =\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/944f08c7a6eadcf5ce6e5bcd7b73894bace891ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.792ex; height:3.009ex;" alt="{\displaystyle \forall x_{i},x_{j}\in S:\langle {x_{i}},{x_{j}}\rangle =\delta _{ij}}"></span>, 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 \delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}"></span> a <a href="/wiki/Kronecker-delta" title="Kronecker-delta">Kronecker-delta</a>. A Zorn-lemmával belátható, hogy minden Hilbert-térnek van ortonormált bázisa. </p><p>Egy véges <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 S=\{x_{n}|n=1,2,...,N\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>N</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\{x_{n}|n=1,2,...,N\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b6b24c6c0e5e35d4a7f793a1265ef0b9c40e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.203ex; height:2.843ex;" alt="{\displaystyle S=\{x_{n}|n=1,2,...,N\}}"></span> ortonormált rendszerre érvényes a <a href="/wiki/Pitagorasz-t%C3%A9tel" title="Pitagorasz-tétel">Pitagorasz-tétel</a> és a <a href="/wiki/Bessel-egyenl%C5%91tlens%C3%A9g" title="Bessel-egyenlőtlenség">Bessel-egyenlőtlenség</a> (mint minden belső szorzatos térben). Azaz minden x-re H-ban: </p><p><b>Pitagorasz:</b> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||x||^{2}=\sum _{n=1}^{N}|\langle {x_{n}},{x}\rangle |^{2}+||x-\sum _{n=1}^{N}\langle {x_{n}},{x}\rangle 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> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <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 ||x||^{2}=\sum _{n=1}^{N}|\langle {x_{n}},{x}\rangle |^{2}+||x-\sum _{n=1}^{N}\langle {x_{n}},{x}\rangle x_{n}||}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130833d5d0bfc11385a13f99842bc8d70c5136e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.103ex; height:7.343ex;" alt="{\displaystyle ||x||^{2}=\sum _{n=1}^{N}|\langle {x_{n}},{x}\rangle |^{2}+||x-\sum _{n=1}^{N}\langle {x_{n}},{x}\rangle x_{n}||}"></span> </p><p><b>Bessel:</b> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||x||^{2}\geq \sum _{n=1}^{N}|\langle {x_{n}},{x}\rangle |^{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> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||x||^{2}\geq \sum _{n=1}^{N}|\langle {x_{n}},{x}\rangle |^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36a7cd859f86f6bd27f7abfc49218dc33268fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.881ex; height:7.343ex;" alt="{\displaystyle ||x||^{2}\geq \sum _{n=1}^{N}|\langle {x_{n}},{x}\rangle |^{2}}"></span> </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Bázis"><span id="B.C3.A1zis"></span>Bázis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=5" title="Szakasz szerkesztése: Bázis"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Definíció:</i> A <i>H</i> Hilbert-tér egy maximális ortonormált rendszerét <b>ortonormált bázis</b>nak nevezzük. Azaz <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\subset H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊂</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subset H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73f0d480f6969eeb6816dbedaeab311b5d2b9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.926ex; height:2.176ex;" alt="{\displaystyle B\subset H}"></span> egy ortonormált bázis, ha <i>B</i> ortonormált rendszer, és <i>B</i> bármely <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\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0957496d2596a81d84e50252c806c5ae488396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.234ex; height:2.176ex;" alt="{\displaystyle x\in H}"></span>-val való bővítés után, már nem ortonormált rendszer. </p><p>A <a href="/wiki/Zorn-lemma" title="Zorn-lemma">Zorn-lemma</a> (illetve a <a href="/wiki/Kiv%C3%A1laszt%C3%A1si_axi%C3%B3ma" title="Kiválasztási axióma">kiválasztási axióma</a>) használatával megmutatható, hogy minden Hilbert-térnek van ortonormált bázisa. </p><p>Ha <i>y</i> egy <i>H</i> Hilbert-térbéli vektor 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 B=\{x_{i}\in H:i\in I\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>H</mi> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{x_{i}\in H:i\in I\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c0e8a91208c4708a6da993c9ab3ab2a8464ea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.973ex; height:2.843ex;" alt="{\displaystyle B=\{x_{i}\in H:i\in I\}}"></span> egy ortonormált bázisa <i>H</i>-nak, ahol <i>I</i> egy tetszőleges indexhalmaz, akkor: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\sum _{i\in I}\langle {x_{i}},{y}\rangle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\sum _{i\in I}\langle {x_{i}},{y}\rangle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0958e0471097e2509f52d155344accbfc5c7df52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.866ex; height:5.676ex;" alt="{\displaystyle y=\sum _{i\in I}\langle {x_{i}},{y}\rangle x_{i}}"></span>, 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 \langle {x_{i}},{y}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {x_{i}},{y}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5962882dd7392164bbeb280f66175a881ba2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.128ex; height:2.843ex;" alt="{\displaystyle \langle {x_{i}},{y}\rangle }"></span> csak <a href="/wiki/Sz%C3%A1moss%C3%A1g" title="Számosság">megszámlálható</a> sok <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 i\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}"></span>-re nem nulla, és az összegzés független a sorrendtől. y kifejezése bázisvektorok <a href="/w/index.php?title=Sor_(matematika)&action=edit&redlink=1" class="new" title="Sor (matematika) (a lap nem létezik)">soraként</a> egyértelmű. Továbbá: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||y||^{2}=\sum _{i\in I}|\langle {x_{i}},{y}\rangle |^{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> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||y||^{2}=\sum _{i\in I}|\langle {x_{i}},{y}\rangle |^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9082cd25a0e9d091071a718ce5e463d4b08c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.114ex; height:5.676ex;" alt="{\displaystyle ||y||^{2}=\sum _{i\in I}|\langle {x_{i}},{y}\rangle |^{2}}"></span> (<b><a href="/w/index.php?title=Parseval_t%C3%A9tel&action=edit&redlink=1" class="new" title="Parseval tétel (a lap nem létezik)">Parseval tétel</a></b>). </p><p>Ortonormált bázisokkal a Hilbert-terek teljesen osztályozhatók. Minden Hilbert-térben van ortonormált bázis, és egy Hilbert-tér ortonormált bázisainak kardinalitása megegyezik. Egy Hilbert-tér ortonormált bázisainak kardinalitása tehát jóldefiniált. Ezt nevezzük a tér Hilbert-dimenziójának, röviden dimenziójának. Ugyanazon test fölötti megegyező dimenziójú Hilbert-terek izomorfak, ugyanis a két bázis elemről elemre megfeleltethető egymásnak, és ez a megfeleltetés folytonosan kiterjeszthető az egész térre. </p> <div class="mw-heading mw-heading2"><h2 id="Alterek">Alterek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=6" title="Szakasz szerkesztése: Alterek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Egy Hilbert-altér egy Hilbert-tér olyan részhalmaza, ami a Hilbert-térben értelmezett vektorösszeadás, skalárral szorzás és skalárszorzás leszűkítésére szintén Hilbert-tér. Ez azt is jelenti, hogy altere, mint vektortérnek, hiszen ezek a kikötések feltételezik a nullvektor tartalmazását, és zárt a vektorösszeadásra és a skalárral szorzásra. Emellett még a skalárszorzásra is teljesnek kell lennie; ez ekvivalens azzal, hogy topológiai értelemben zárt. Emiatt a Hilbert-altereket zárt alterekként is emlegetik, szemben az egyszerűen csak altérként említett vektorterekkel. Általában ezek az alterek skalárszorzatos vektorterek, melyek sűrűek egy Hilbert-térben, ami lezárással kapható. Lehetséges Hilbert-alterekre hányadosteret képezni, ekkor szintén Hilbert-térhez jutunk. </p><p>Ez hasonló a Banach-terek esetéhez, melyek vektortéri értelemben vett alterei normált terek. Egy fontos különbség a projekciós tétel: Adva legyen egy Hilbert-tér, amiben kiválasztunk egy elemet, és egy Hilbert-alteret. Ekkor a Hilbert-altérben egyértelműen van egy vektor, melynek az adott vektortól mért távolsága minimális. Banach-terekben ez általában már véges dimenzióban sem igaz. Ez lehetővé teszi Hilbert-altér hányadosterének kanonikus azonosítását egy Hilbert-altérrel, ez az ortogonális komplementer; és az ortogonális vetítés bevezetését is. Egy Hilbert-altér ortogonális komplementere egy komplementer Hilbert-altér; azonban Banach-alterekhez általában nincs komplementer altér. </p><p><i>Definíció:</i> 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 S\subset H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b29f290397b3868fcdb875df4cbc7840a5314a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.661ex; height:2.176ex;" alt="{\displaystyle S\subset H}"></span>, ekkor definiáljuk S ortogonális komplementerét: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\bot }:=\{x\in H|\langle {x},{y}\rangle =0\quad \forall y\in S\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{\bot }:=\{x\in H|\langle {x},{y}\rangle =0\quad \forall y\in S\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754bbcd1588e4fb3d490f9974b5dfb2453ecb82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.683ex; height:3.176ex;" alt="{\displaystyle S^{\bot }:=\{x\in H|\langle {x},{y}\rangle =0\quad \forall y\in S\}}"></span>. </p><p><i>Tétel</i>: Legyen <i>H</i> egy Hilbert-tér, <i>M</i> pedig egy <a href="/wiki/Topol%C3%B3gia" title="Topológia">zárt</a> <a href="/wiki/Line%C3%A1ris_alt%C3%A9r" title="Lineáris altér">altér</a> <i>H</i>-ban. 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 H=M\oplus M^{\bot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>M</mi> <mo>⊕</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=M\oplus M^{\bot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc58326c60915e1a7488f8def712cb40a560d482" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.454ex; height:2.843ex;" alt="{\displaystyle H=M\oplus M^{\bot }}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Konjugált_Hilbert-tér"><span id="Konjug.C3.A1lt_Hilbert-t.C3.A9r"></span>Konjugált Hilbert-tér</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=7" title="Szakasz szerkesztése: Konjugált Hilbert-tér"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Komplex Hilbert-terek esetén a skalárszorzás nem szimmetrikus; lineáris a második argumentumban, és szemilineáris az elsőben. Azonban definiálható a konjugált Hilbert-tér, a következőképpen: 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> Hilbert-tér, és legyen 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 {\overline {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e3c5eac166e464406970ed2cadc14fa7345da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.265ex; height:3.009ex;" alt="{\displaystyle {\overline {H}}}"></span> értelmezve ugyanazon az alaphalmazon, és legyen a vektorok összeadása is ugyanaz, mint <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>-ban. A többi művelet: </p> <ul><li>Skalárral szorzás: <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 \lambda \cdot _{\overline {H}}u:={\overline {\lambda }}u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> </msub> <mi>u</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \cdot _{\overline {H}}u:={\overline {\lambda }}u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9ecd82918bc680b1fa070ec0b911c04cb5859c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.755ex; height:3.843ex;" alt="{\displaystyle \lambda \cdot _{\overline {H}}u:={\overline {\lambda }}u}"></span></li> <li>Skalárszorzás: <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 \langle u,v\rangle _{\overline {H}}:={\overline {\langle u,v\rangle }}=\langle v,u\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle _{\overline {H}}:={\overline {\langle u,v\rangle }}=\langle v,u\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e129d5bb4248343611da0ba124e3c8c05532481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.706ex; height:4.009ex;" alt="{\displaystyle \langle u,v\rangle _{\overline {H}}:={\overline {\langle u,v\rangle }}=\langle v,u\rangle }"></span>.</li></ul> <p>Ezekkel a műveletekkel <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 {\overline {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e3c5eac166e464406970ed2cadc14fa7345da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.265ex; height:3.009ex;" alt="{\displaystyle {\overline {H}}}"></span> szintén Hilbert-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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> konjugált Hilbert-tere. A konjugált Hilbert-tér konjugált Hilbert-tere, az eredeti Hilbert-tér. </p> <div class="mw-heading mw-heading2"><h2 id="Hilbert-terek_közötti_leképezések"><span id="Hilbert-terek_k.C3.B6z.C3.B6tti_lek.C3.A9pez.C3.A9sek"></span>Hilbert-terek közötti leképezések</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=8" title="Szakasz szerkesztése: Hilbert-terek közötti leképezések"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A funkcionálanalízisben vizsgálnak olyan terek közötti leképezéseket is, amelyek megtartják a terek struktúráját. Ezek a leképezések megtartják a vektortér struktúrát is, azaz lineáris leképezések, melyeket a funkcionálanalízisben lineáris operátoroknak neveznek. </p><p>A Hilbert-terek közötti lineáris operátorok fontos osztálya a folytonos lineáris operátoroké. Ezek megtartják a topologikus struktúrát, így a konvergenciát is. További fontos tulajdonságok valamilyen értelmű korlátosságot feltételeznek. A korlátosság ekvivalens a folytonossággal; így sokszor egyszerűen csak folytonos operátorokként emlegetik őket. A kompaktság egy erősebb követelmény. A Schatten-Neumann-osztályok a kompakt operátorok osztályának valódi alosztályai. Az operátorok osztályain szintén definiálnak normákat és operátortopológiákat. </p><p>Az unitér operátorok a Hilbert-terek természetes izomorfizmus fogalmát definiálják, mivel ezek éppen az izomorfizmusok a Hilbert-terek kategóriájában, a skalárszorzattartó lineáris leképezésekkel, mint morfizmusokkal. Ezek konkrétan a lineáris szürjektív izometriák, a szögek és hosszak megőrzésével. </p><p>A folytonos lineáris operátorokat meghatározza, hogy egy ortonormált bázist mire képez le. Valójában minden kardinális számhoz létezik valós és komplex Hilbert-tér is, melynek a dimenziója megegyezik az adott kardinális számmal, például az <a href="/w/index.php?title=%E2%84%932-t%C3%A9r&action=edit&redlink=1" class="new" title="ℓ2-tér (a lap nem létezik)"><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 \ell ^{2}(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da952c66063a82ac4fb47473accc8669da064622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(I)}"></span></a> tér, ahol 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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> indexhalmaz kardinális száma az adott kardinális szám: </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 \ell ^{2}(I):=\left\{u\colon I\to K\mid \sum _{i\in I}\left|u(i)\right|^{2}<\infty \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <mi>u</mi> <mo>:</mo> <mi>I</mi> <mo stretchy="false">→</mo> <mi>K</mi> <mo>∣</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(I):=\left\{u\colon I\to K\mid \sum _{i\in I}\left|u(i)\right|^{2}<\infty \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a88f8bec46267399739cdcc3d981ce924208a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.103ex; height:7.509ex;" alt="{\displaystyle \ell ^{2}(I):=\left\{u\colon I\to K\mid \sum _{i\in I}\left|u(i)\right|^{2}<\infty \right\}}"></span>,</dd></dl> <p>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 K=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }"></span> vagy <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=\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58665cdd4df26adaa88a248908d1481041a77c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {C} }"></span>, és a konvergencia érdekében előírjuk, hogy csak megszámlálható sok tag különbözik a nullától (lásd feltétlen kovergencia). Ezt a teret ellátjuk az </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 \langle u,v\rangle :=\sum _{i\in I}{\overline {u(i)}}v(i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle :=\sum _{i\in I}{\overline {u(i)}}v(i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5532208cdb7c65fd341010088bfc4fcb097e4b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.584ex; height:6.009ex;" alt="{\displaystyle \langle u,v\rangle :=\sum _{i\in I}{\overline {u(i)}}v(i)}"></span>,</dd></dl> <p>skalárszorzattal. 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 \ell ^{2}(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da952c66063a82ac4fb47473accc8669da064622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(I)}"></span> tér egy ortonormált bázisát alkotják 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 u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span> vektorok, 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 u_{i}(j)=\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}(j)=\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811559a49fc57fb1dff8961972702a0dac5a8d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.505ex; height:3.009ex;" alt="{\displaystyle u_{i}(j)=\delta _{ij}}"></span>. A Riesz-Fischer-tétel azt mondja ki, hogy minden Hilbert-tér izomorf egy ilyen térrel. </p> <div class="mw-heading mw-heading2"><h2 id="Riesz_reprezentációs_tétel"><span id="Riesz_reprezent.C3.A1ci.C3.B3s_t.C3.A9tel"></span>Riesz reprezentációs tétel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=9" title="Szakasz szerkesztése: Riesz reprezentációs tétel"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="dablink noprint noviewer" style="padding-left: 2em; vertical-align: middle;" cellpadding="0" cellspacing="0"><tbody><tr><td style="padding-right:.25em;"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Searchtool_right.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/14px-Searchtool_right.svg.png" decoding="async" width="14" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/21px-Searchtool_right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Searchtool_right.svg/28px-Searchtool_right.svg.png 2x" data-file-width="60" data-file-height="60" /></a></span></td><td><i>Bővebben: <a href="/w/index.php?title=Riesz_reprezent%C3%A1ci%C3%B3s_t%C3%A9tel&action=edit&redlink=1" class="new" title="Riesz reprezentációs tétel (a lap nem létezik)">Riesz reprezentációs tétel</a></i></td></tr></tbody></table> <p><i>Definíció (<a href="/wiki/Du%C3%A1lis_t%C3%A9r" title="Duális tér">duális tér</a>):</i> Egy <i>H</i> Hilbert-tér <i>H*</i> duális terén, a <i>H</i>-n értelmezett <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)">folytonos</a> <a href="/wiki/Line%C3%A1ris_lek%C3%A9pez%C3%A9s" title="Lineáris leképezés">lineáris</a> <a href="/wiki/Funkcion%C3%A1l" title="Funkcionál">funkcionálok</a> <a href="/wiki/Banach-t%C3%A9r" title="Banach-tér">Banach-terét</a> értjük, azaz </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}:=\{T:H\rightarrow \mathbb {C} \quad |\quad T\quad {\mbox{lineáris és folytonos}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="1em" /> <mi>T</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>lineáris és folytonos</mtext> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{*}:=\{T:H\rightarrow \mathbb {C} \quad |\quad T\quad {\mbox{lineáris és folytonos}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c97a952302fb6c56eaae0476fafacf21cf4a1f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:49.599ex; height:3.343ex;" alt="{\displaystyle H^{*}:=\{T:H\rightarrow \mathbb {C} \quad |\quad T\quad {\mbox{lineáris és folytonos}}\}}"></span> </p><p>a folytonosság (mivel <a href="/wiki/Norm%C3%A1lt_t%C3%A9r" title="Normált tér">normált terek</a> közötti lineáris leképzésről van szó) egyenértékű a leképzés <a href="/wiki/Oper%C3%A1tornorma" title="Operátornorma">operátornorma</a> szerinti korlátosságával, azaz 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 T:H\rightarrow \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:H\rightarrow \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b991ca0a3dc46f0dfc12d9405588913c137b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.929ex; height:2.176ex;" alt="{\displaystyle T:H\rightarrow \mathbb {C} }"></span> lineáris függvényre igaz: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\quad {\mbox{folytonos}}\qquad \Longleftrightarrow \qquad ||T||<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>folytonos</mtext> </mstyle> </mrow> <mspace width="2em" /> <mo stretchy="false">⟺</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\quad {\mbox{folytonos}}\qquad \Longleftrightarrow \qquad ||T||<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ffc961b6c911da8ada0c10c1a60e0852ce10bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.689ex; height:2.843ex;" alt="{\displaystyle T\quad {\mbox{folytonos}}\qquad \Longleftrightarrow \qquad ||T||<\infty }"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||T||:=\sup\{{\frac {|Tx|}{||x||}}\ |\quad x\neq 0,\quad x\in H\}}"> <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>T</mi> <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"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||T||:=\sup\{{\frac {|Tx|}{||x||}}\ |\quad x\neq 0,\quad x\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ec498426dc3199ae84976f663e468fa3d335a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.622ex; height:6.509ex;" alt="{\displaystyle ||T||:=\sup\{{\frac {|Tx|}{||x||}}\ |\quad x\neq 0,\quad x\in H\}}"></span> </p><p><i>Tétel (Riesz reprezentáció)</i>: 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 T\in H^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>∈</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\in H^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f636f7bf1078ebefb9418d503cfe0a9c3d2ed87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.635ex; height:2.343ex;" alt="{\displaystyle T\in H^{*}}"></span>-hez létezik pontosan 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 y_{T}\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{T}\in H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cfec908fde4a4b14cbb62d89c1cb4f0235713e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.433ex; height:2.509ex;" alt="{\displaystyle y_{T}\in H}"></span>, úgy hogy <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 T(x)=\langle {y_{T}},{x}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)=\langle {y_{T}},{x}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651a13f4b7e49ec3a0ae9724b7e27f38dc2a1c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.575ex; height:2.843ex;" alt="{\displaystyle T(x)=\langle {y_{T}},{x}\rangle }"></span> minden x-re H-ban, és<br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||y_{T}||=||T||}"> <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> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <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> <mi>T</mi> <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 ||y_{T}||=||T||}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d5a49cffdf158713e80676e017b5676ce3103e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.438ex; height:2.843ex;" alt="{\displaystyle ||y_{T}||=||T||}"></span>. </p><p>Vagyis a tétel azt mondja ki, hogy H duális tere egy Hilbert-tér, amely izometrikusan izomorf H-hoz. Ez az egyik leglényegesebb tulajdonsága a Hilbert-tereknek, és ez a tulajdonság különbözteti meg őket nagyban az általánosabb Banach-terektől. Komplex esetben a tétel hasonlóan működik, azzal a különbséggel, hogy a leképezés szemilineáris, tehát az operátor is szemilineáris. Mindkét esetben a Hilbert-tér izomorf a duális terével (egy szemiunitér <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 H\to H^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\to H^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b5076003d0d1003c97338bd7a0bc12b5add0e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.466ex; height:2.509ex;" alt="{\displaystyle H\to H^{\prime }}"></span> operátor felbontható 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 H\to H^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\to H^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b5076003d0d1003c97338bd7a0bc12b5add0e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.466ex; height:2.509ex;" alt="{\displaystyle H\to H^{\prime }}"></span> unitér és 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 H^{\prime }\to H^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{\prime }\to H^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb304787691224c9a499fe8fe68aff6779b39836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.19ex; height:2.509ex;" alt="{\displaystyle H^{\prime }\to H^{\prime }}"></span> szemiunitér operátorra), így a Hilbert-tér izomorf a biduális terével, tehát a Hilbert-terek reflexívek. </p><p>Ezen tétel felhasználásával vezetik be a fizikusok a bra-ket írásmódot, mely a Hilbert-tér elemeit <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\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48004887d8f9dfc489bd2bc793780b7f1d8039ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.881ex; height:2.843ex;" alt="{\displaystyle |x\rangle }"></span> módon jelöli, és ket-vektoroknak nevezi őket, a duálvektorokat pedig <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 \langle x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60c60e3538943708f60e14a5a7f4a1beb076af7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.881ex; height:2.843ex;" alt="{\displaystyle \langle x|}"></span> módon, melyeket bra-vektoroknak nevez. Két vektor skaláris szorzata, pedig a duálvektor hattatása a vektorra: <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 \langle y|(|x\rangle )=\langle {y}|{x}\rangle =\langle {y},{x}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle y|(|x\rangle )=\langle {y}|{x}\rangle =\langle {y},{x}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2556a22c1bdb170e9a58346e165016697b661c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.864ex; height:2.843ex;" alt="{\displaystyle \langle y|(|x\rangle )=\langle {y}|{x}\rangle =\langle {y},{x}\rangle }"></span>, azaz a duálvektort a vektor mellé írjuk, így a bra és a ket vektor képzi nyelvi humorral a bracket-et, azaz a skaláris szorzat jelölésére használt zárójelet. </p><p>A tételből következik, hogy 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}"> <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>-ből <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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>-ba menő lineáris operátor adjungált operátora értelmezhető, mint 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 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>-ból <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>-be menő lineáris operátor. Így egy operátor felcserélhető adjungált operátorával; az efféle operátorok alkotják a normális operátorok osztályát. Egy Hilbert-tér operátorainál fennáll annak a lehetősége, hogy egy operátor adjungált operátora önmaga. Ezek az önadjungált operátorok. </p><p>Egy Hilbert-téren több fent bevezetett operátorosztály operátoralgebrát alkot. Az adjungálással, mint involúcióval és egy megfelelő normával involutív Banach-algebrákat alkotnak. Egy Hilbert-tér folytonos lineáris operátorai az adjungálással és az operátornormával C*-algebrát alkot. </p> <div class="mw-heading mw-heading2"><h2 id="Fourier-együttható"><span id="Fourier-egy.C3.BCtthat.C3.B3"></span>Fourier-együttható</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=10" title="Szakasz szerkesztése: Fourier-együttható"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az ortonormált bázisok hasznosak a Hilbert-terek és elemeik vizsgálatára mind valós, mind komplex test fölött. Például az elemek ábrázolása meghatározható ortonormált bázisban. 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 B=(b_{1},b_{2},\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(b_{1},b_{2},\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abaa29f970d0b849d6389e14f3e35945f5c6fe73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.566ex; height:2.843ex;" alt="{\displaystyle B=(b_{1},b_{2},\dots )}"></span> ortonormált bázis, 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}"> <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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> a Hilbert-tér egy vektora. Mivel <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> ortonormált bázis, azért 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 \alpha _{k}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{k}\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246a49c0e92a0618f27d613e7b70f36a1cb1f386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.095ex; height:2.509ex;" alt="{\displaystyle \alpha _{k}\in \mathbb {R} }"></span> illetve <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> együtthatók úgy, hogy </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 v=\sum _{k}\alpha _{k}b_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\sum _{k}\alpha _{k}b_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565e8dee8237e98bb0707c49e6486ea59ca4e742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.631ex; height:5.509ex;" alt="{\displaystyle v=\sum _{k}\alpha _{k}b_{k}}"></span>.</dd></dl> <p>Ezek az együtthatók meghatározhatók az ortonormált bázis speciális tulajdonságainak felhasználásával </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 \langle b_{n},v\rangle =\left\langle b_{n},\sum _{k}\alpha _{k}b_{k}\right\rangle =\sum _{k}\alpha _{k}\langle b_{n},b_{k}\rangle =\alpha _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle b_{n},v\rangle =\left\langle b_{n},\sum _{k}\alpha _{k}b_{k}\right\rangle =\sum _{k}\alpha _{k}\langle b_{n},b_{k}\rangle =\alpha _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbbb2075deed07400ee531f91501c6c9e928c233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.053ex; height:7.509ex;" alt="{\displaystyle \langle b_{n},v\rangle =\left\langle b_{n},\sum _{k}\alpha _{k}b_{k}\right\rangle =\sum _{k}\alpha _{k}\langle b_{n},b_{k}\rangle =\alpha _{n}}"></span>,</dd></dl> <p>mivel a különböző bázisvektorok skalárszorzata nulla, és a bázisvektorok önmagukkal vett skalárszorzata 1. Egy vektor ábrázolásának <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-edik együtthatója ortonormált bázisban meghatározható skalárszorzattal. Ezeket az együtthatókat Fourier-együtthatóknak is nevezzük, mivel a Fourier-analízis egy fogalmának általánosítását nyújtják. </p> <div class="mw-heading mw-heading2"><h2 id="RKHS">RKHS</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=11" title="Szakasz szerkesztése: RKHS"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ha egy Hilbert-teret egy maggal asszociálunk, melyet a térben minden függvény reprodukál, akkor Reproducing Kernel Hilbert Space (RKHS)-ről van szó, ami magyarra fordítva: reprodukáló mag Hilbert-tér. Ezt először Stanisław Zaremba matematikus formalizálta 1907-ben. Jelentősége fél évszázaddal később nőtt meg, amikor a funkcionálanalízisben fontos szerephez jutott. Ma a reprodukáló magos Hilbert-terek a statisztikai elméletek egy szokványos eszköze, különösen a gépi tanulásban. </p> <div class="mw-heading mw-heading2"><h2 id="Alkalmazások"><span id="Alkalmaz.C3.A1sok"></span>Alkalmazások</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=12" title="Szakasz szerkesztése: Alkalmazások"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Minden Hilbert-tér egyben <a href="/wiki/Banach-t%C3%A9r" title="Banach-tér">Banach-tér</a> is (de fordítva nem igaz).<br /> Minden <a href="/w/index.php?title=L-2_t%C3%A9r&action=edit&redlink=1" class="new" title="L-2 tér (a lap nem létezik)">L-2 tér</a> egy Hilbert-tér. </p><p>Minden véges dimenziós belső szorzattal rendelkező tér (mint az <a href="/wiki/Euklideszi_t%C3%A9r_(line%C3%A1ris_algebra)" title="Euklideszi tér (lineáris algebra)">Euklideszi-tér</a> a szokásos skalárszorzattal) Hilbert-teret alkot. Valójában a végtelen dimenziós terek jelentősége az alkalmazások területén sokkal nagyobb. Pár példa ezekre: </p> <ul><li>Az <a href="/w/index.php?title=Unit%C3%A9r_reprezent%C3%A1ci%C3%B3&action=edit&redlink=1" class="new" title="Unitér reprezentáció (a lap nem létezik)">unitér csoportreprezentációk</a> elmélete</li> <li>A négyzetesen integrálható <a href="/wiki/Sztochasztikus_folyamat" title="Sztochasztikus folyamat">sztochasztikus folyamatok</a></li> <li>A <a href="/wiki/Parci%C3%A1lis_differenci%C3%A1legyenlet" title="Parciális differenciálegyenlet">parciális differenciálegyenletek</a> Hilbert-tér elmélete, különösen a <a href="/w/index.php?title=Dirichlet-probl%C3%A9ma&action=edit&redlink=1" class="new" title="Dirichlet-probléma (a lap nem létezik)">Dirichlet-probléma</a> megfogalmazásai. Lásd még: <a href="/w/index.php?title=A_parci%C3%A1lis_differenci%C3%A1legyenletek_megold%C3%A1selm%C3%A9lete&action=edit&redlink=1" class="new" title="A parciális differenciálegyenletek megoldáselmélete (a lap nem létezik)">a parciális differenciálegyenletek megoldáselmélete</a>, amitől a Hilbert-terek a fizikában is nagy fontosságot nyernek</li> <li>A függvények spektrális analízise, beleértve a <a href="/w/index.php?title=Wavelet&action=edit&redlink=1" class="new" title="Wavelet (a lap nem létezik)">waveleteket</a></li></ul> <p>A <a href="/wiki/Kvantummechanika" title="Kvantummechanika">kvantummechanika</a> matematikai megfogalmazásai. Például a <a href="/wiki/Kvantummechanika" title="Kvantummechanika">kvantummechanikában</a> egy kvantummechanikai rendszer <a href="/w/index.php?title=Tiszta_%C3%A1llapot&action=edit&redlink=1" class="new" title="Tiszta állapot (a lap nem létezik)">tiszta állapotai</a> megadhatók Hilbert-térben egy vektorral. Egy kvantummechanikai rendszer állapotai egy lineáris struktúra elemei, vagyis állapotok lineáris kombinációja szintén állapot. Két állapot, <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 |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></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 |\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span> skalárszorzatának normájának négyzete azt adja meg, hogy ha egy mérés eredménye <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 |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span>, akkor mekkora annak a valószínűsége, hogy a rendszer 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 |\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span> állapotban van. Ha a fizikában a Hilbert-térről beszélnek, akkor az adott kvantummechanikai rendszer állapotterét értik. </p><p>Például egy szabad részecske lehetséges hullámfüggvényei 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 \psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b69d6e411b25255fc42ffdc3afe3a29f42b8ee20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.572ex; height:3.009ex;" alt="{\displaystyle \psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {C} }"></span> négyzetesen integrálható függvények <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"></span> terét alkotják a szokásos <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"></span>-skalárszorzattal: </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 \textstyle \langle \psi \,|\,\phi \rangle =\int _{\mathbb {R} ^{3}}\psi ^{*}({\vec {x}})\,\phi ({\vec {x}})\,{\rm {d}}{\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msub> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <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> <mspace width="thinmathspace" /> <mi>ϕ</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 stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \langle \psi \,|\,\phi \rangle =\int _{\mathbb {R} ^{3}}\psi ^{*}({\vec {x}})\,\phi ({\vec {x}})\,{\rm {d}}{\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b517ac529ff2f459a547e0c45b13924520cc24ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.589ex; height:3.176ex;" alt="{\displaystyle \textstyle \langle \psi \,|\,\phi \rangle =\int _{\mathbb {R} ^{3}}\psi ^{*}({\vec {x}})\,\phi ({\vec {x}})\,{\rm {d}}{\vec {x}}}"></span>.</dd></dl> <p>Egy másik példa egy elektron lehetséges spin állapotai 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 {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}"></span> teret feszítik ki, a szokásos komplex skalárszorzattal. </p><p>A belső szorzat teszi lehetővé a „geometriai” látásmód megőrzését, és a véges dimenziós terekben megszokott geometriai nyelvezet használatát. Az összes végtelen dimenziós <a href="/w/index.php?title=Topologikus_vektort%C3%A9r&action=edit&redlink=1" class="new" title="Topologikus vektortér (a lap nem létezik)">topologikus vektortér</a> közül a Hilbert-terek a „legjobban viselkedőek” és ezek állnak legközelebb a véges dimenziós terekhez. A funkcionálanalízis szempontjából a Hilbert-terek speciális és egyszerű szerkezetű terek egy osztályát alkotják. </p><p>A <a href="/wiki/Fourier-anal%C3%ADzis" title="Fourier-analízis">Fourier-analízis</a> egyik célja, hogy egy adott függvényt adott alapfüggvények kombinációjaként írjunk fel, azaz olyan (esetleg végtelen) összegként, melyben az alapfüggvények többszörösei a tagok. Ez a probléma absztrakt módon vizsgálható Hilbert-terekben: minden Hilbert-térnek van <a href="/wiki/Ortonorm%C3%A1lt_b%C3%A1zis" title="Ortonormált bázis">ortonormált bázisa</a>, és a Hilbert-tér minden eleme egyféleképp írható fel a báziselemek kombinációjaként, azaz olyan összegként, melyben a bázisvektorok többszörösei (skalárszorosai) szerepelnek. </p> <div class="mw-heading mw-heading2"><h2 id="Érdekesség"><span id=".C3.89rdekess.C3.A9g"></span>Érdekesség</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=13" title="Szakasz szerkesztése: Érdekesség"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A német nyelvterületen több egyetemen is van Hilbert-térnek nevezett terem.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<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=Hilbert-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">Simonovits András: Válogatott fejezetek a matematika történetéből. 146-148. old. Typotex Kiadó, 2009. <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9789632790268" title="Speciális:Könyvforrások/9789632790268">ISBN 978-963-279-026-8</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"> <span class="citation"><a rel="nofollow" class="external text" href="https://www.mathematik.uni-konstanz.de/fachschaft/ueber-uns/"><i>Hilbertraum der Fachschaft Mathematik an der Universität Konstanz</i></a></span> </span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20191008112456/https://www.mathematik.uni-mainz.de/mathematik-und-schule/"><i>Freunde der Mathematik an der Johannes Gutenberg-Universität Mainz, Veranstaltung Mathematik und Schule</i></a>. [2019. október 8-i dátummal az <a rel="nofollow" class="external text" href="https://www.mathematik.uni-mainz.de/mathematik-und-schule/">eredetiből</a> archiválva]. (Hozzáférés: 2024. május 19.)</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"> <span class="citation"><a rel="nofollow" class="external text" href="http://fachschaft-physik.tu-dortmund.de/wordpress/studium/raumlichkeiten/"><i>Hilbertraum der Fachschaft Physik an der Technischen Universität Dortmund</i></a></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="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=Hilbert-t%C3%A9r&action=edit&section=15" 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/Hilbertraum" class="extiw" title="de:Hilbertraum">Hilbertraum</a></i> című német Wikipédia-szócikk fordításán alapul. Az eredeti cikk szerkesztőit annak laptörténete sorolja fel. Ez a jelzés csupán a megfogalmazás eredetét és a szerzői jogokat jelzi, nem szolgál a cikkben szereplő információk forrásmegjelöléseként. </p> <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=Hilbert-t%C3%A9r&action=edit&section=16" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><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, 2005, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/3540435867" title="Speciális:Könyvforrások/3540435867">ISBN 3-540-43586-7</a>, Kapitel V, VI und VII.</li> <li><a href="/w/index.php?title=Richard_Kadison&action=edit&redlink=1" class="new" title="Richard Kadison (a lap nem létezik)">Richard V. Kadison</a>, <a href="/w/index.php?title=John_R._Ringrose&action=edit&redlink=1" class="new" title="John R. Ringrose (a lap nem létezik)">John R. Ringrose</a>: <i>Fundamentals of the Theory of Operator Algebras</i>, Band 1: <i>Elementary Theory</i>; Academic Press, New York NY, 1983, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/0123933013" title="Speciális:Könyvforrások/0123933013">ISBN 0-12-393301-3</a> (<i>Pure and Applied Mathematics</i> 100, 1), Chapter 2: <i>Basics of Hilbert Space and Linear Operators</i></li></ul> <div class="mw-heading mw-heading2"><h2 id="További_információk"><span id="Tov.C3.A1bbi_inform.C3.A1ci.C3.B3k"></span>További információk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbert-t%C3%A9r&action=edit&section=17" title="Szakasz szerkesztése: További információk"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Michael Reed, Barry Simon: Functional Analysis (Methods of Modern Mathematical Physics, Volume 1), 1980</li></ul> <p><br /> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r26593303">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r26641489">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{width:100%;line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><style data-mw-deduplicate="TemplateStyles:r26643308">@media screen and (max-width:719px){.mw-parser-output div.navbox.authoritycontrol{display:block}.mw-parser-output .authoritycontrol tbody,.mw-parser-output .authoritycontrol tr,.mw-parser-output .authoritycontrol th,.mw-parser-output .authoritycontrol td,.mw-parser-output .authoritycontrol .navbox-row>th+td{display:block;text-align:center}.mw-parser-output .authoritycontrol .navbox-list-with-group{border:none}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26593303"></div><div role="navigation" class="navbox authoritycontrol" aria-labelledby="Nemzetközi_katalógusok" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Nemzetközi_katalógusok" scope="row" class="navbox-group" style="width:auto"><a href="/wiki/Sablon:Nemzetk%C3%B6zi_katal%C3%B3gusok/doc" title="Sablon:Nemzetközi katalógusok/doc">Nemzetközi katalógusok</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kongresszusi_K%C3%B6nyvt%C3%A1r" title="Kongresszusi Könyvtár">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://lccn.loc.gov/sh85060803">sh85060803</a></span></li> <li><a href="/wiki/Integr%C3%A1lt_katal%C3%B3gust%C3%A1r" title="Integrált katalógustár">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4159850-7">4159850-7</a></span></li> <li><a href="/wiki/A_Cseh_K%C3%B6zt%C3%A1rsas%C3%A1g_Nemzeti_K%C3%B6nyvt%C3%A1ra" title="A Cseh Köztársaság Nemzeti Könyvtára">NKCS</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph117602&CON_LNG=ENG">ph117602</a></span></li> <li><a href="/wiki/Francia_Nemzeti_K%C3%B6nyvt%C3%A1r" title="Francia Nemzeti Könyvtár">BNF</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11979628h">cb11979628h</a></span></li> <li><a href="/wiki/Spanyol_Nemzeti_K%C3%B6nyvt%C3%A1r" title="Spanyol Nemzeti Könyvtár">BNE</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX531621">XX531621</a></span></li> <li><a href="/wiki/Orsz%C3%A1ggy%C5%B1l%C3%A9si_K%C3%B6nyvt%C3%A1r_(Jap%C3%A1n)" title="Országgyűlési Könyvtár (Japán)">KKT</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00563198">00563198</a></span></li></ul> </div></td></tr></tbody></table></div> <div class="noprint noviewer" style="overflow: hidden; clear: both;"><div style="margin-left:0; margin-right:2px;"><ul style="display:block; list-style-image:none; list-style-type:none; width:100%; vertical-align:middle; margin:0; padding:0; min-height: 27px;"><li style="float:left; min-height: 27px; line-height:25px; width:100%; margin:0; margin-top:.5em; margin-left:0; margin-right:0; padding:0; border:1px solid #CCF; background-color:#F0EEFF"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:P_cartesian_graph.svg" class="mw-file-description" title="Matematika"><img alt="Matematika" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/25px-P_cartesian_graph.svg.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/38px-P_cartesian_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/P_cartesian_graph.svg/50px-P_cartesian_graph.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> <b><a href="/wiki/Port%C3%A1l:Matematika" title="Portál:Matematika">Matematikaportál</a></b> • összefoglaló, színes tartalomajánló lap</li></ul></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">A lap eredeti címe: „<a dir="ltr" href="https://hu.wikipedia.org/w/index.php?title=Hilbert-tér&oldid=27407869">https://hu.wikipedia.org/w/index.php?title=Hilbert-tér&oldid=27407869</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikip%C3%A9dia:Kateg%C3%B3ri%C3%A1k" title="Wikipédia:Kategóriák">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Anal%C3%ADzis" title="Kategória:Analízis">Analízis</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Rejtett kategóriák: <ul><li><a href="/wiki/Kateg%C3%B3ria:Wikip%C3%A9dia-sz%C3%B3cikkek_LCCN-azonos%C3%ADt%C3%B3val" title="Kategória:Wikipédia-szócikkek LCCN-azonosítóval">Wikipédia-szócikkek LCCN-azonosítóval</a></li><li><a href="/wiki/Kateg%C3%B3ria:Wikip%C3%A9dia-sz%C3%B3cikkek_GND-azonos%C3%ADt%C3%B3val" title="Kategória:Wikipédia-szócikkek GND-azonosítóval">Wikipédia-szócikkek GND-azonosítóval</a></li><li><a href="/wiki/Kateg%C3%B3ria:Wikip%C3%A9dia-sz%C3%B3cikkek_BNF-azonos%C3%ADt%C3%B3val" title="Kategória:Wikipédia-szócikkek BNF-azonosítóval">Wikipédia-szócikkek BNF-azonosítóval</a></li><li><a href="/wiki/Kateg%C3%B3ria:Wikip%C3%A9dia-sz%C3%B3cikkek_KKT-azonos%C3%ADt%C3%B3val" title="Kategória:Wikipédia-szócikkek KKT-azonosítóval">Wikipédia-szócikkek KKT-azonosítóval</a></li><li><a href="/wiki/Kateg%C3%B3ria:Lapok_elavult_form%C3%A1tum%C3%BA_matematikai_tagekkel" title="Kategória:Lapok elavult formátumú matematikai tagekkel">Lapok elavult formátumú matematikai tagekkel</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> A lap utolsó módosítása: 2024. augusztus 30., 15:24</li> <li id="footer-info-copyright">A lap szövege <a rel="nofollow" class="external text" href="http://creativecommons.org/licenses/by-sa/4.0/deed.hu">Creative Commons Nevezd meg! – Így add tovább! 4.0</a> licenc alatt van; egyes esetekben más módon is felhasználható. Részletekért lásd a <a href="/wiki/Wikip%C3%A9dia:Felhaszn%C3%A1l%C3%A1si_felt%C3%A9telek" title="Wikipédia:Felhasználási feltételek">felhasználási feltételeket</a>.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Adatvédelmi irányelvek</a></li> <li id="footer-places-about"><a href="/wiki/Wikip%C3%A9dia:R%C3%B3lunk">A Wikipédiáról</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikip%C3%A9dia:Jogi_nyilatkozat">Jogi nyilatkozat</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Magatartási kódex</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Fejlesztők</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/hu.wikipedia.org">Statisztikák</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Sütinyilatkozat</a></li> <li id="footer-places-mobileview"><a href="//hu.m.wikipedia.org/w/index.php?title=Hilbert-t%C3%A9r&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobil nézet</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5857dfdcd6-5d5cn","wgBackendResponseTime":255,"wgPageParseReport":{"limitreport":{"cputime":"0.262","walltime":"0.443","ppvisitednodes":{"value":1453,"limit":1000000},"postexpandincludesize":{"value":11848,"limit":2097152},"templateargumentsize":{"value":2494,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":13597,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 211.502 1 -total"," 50.47% 106.753 1 Sablon:Nemzetközi_katalógusok"," 27.12% 57.365 1 Sablon:Jegyzetek"," 26.21% 55.429 2 Sablon:References"," 11.95% 25.271 3 Sablon:ISBN"," 10.43% 22.057 3 Sablon:Cite_web"," 8.15% 17.243 1 Sablon:Fordítás"," 5.94% 12.561 2 Sablon:Isodátum"," 5.92% 12.518 1 Sablon:Portál"," 4.46% 9.424 1 Sablon:A(z)"]},"scribunto":{"limitreport-timeusage":{"value":"0.085","limit":"10.000"},"limitreport-memusage":{"value":1466004,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-6d555f5f66-cddcq","timestamp":"20241202104643","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Hilbert-t\u00e9r","url":"https:\/\/hu.wikipedia.org\/wiki\/Hilbert-t%C3%A9r","sameAs":"http:\/\/www.wikidata.org\/entity\/Q190056","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q190056","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-10-17T12:48:47Z"}</script> </body> </html>

CINXE.COM

Hilbert-tér – Wikipédia