CINXE.COM

<!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-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Hilbert space - Wikipedia</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-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);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":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"dccda238-d79f-42e7-8c3a-d565d5e6e9b8","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Hilbert_space","wgTitle":"Hilbert space","wgCurRevisionId":1255844830,"wgRevisionId":1255844830,"wgArticleId":20598932,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Pages displaying wikidata descriptions as a fallback via Module:Annotated link","Commons category link from Wikidata","Good articles","Hilbert spaces","Functional analysis","Linear algebra","Operator theory","David Hilbert"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Hilbert_space","wgRelevantArticleId":20598932,"wgIsProbablyEditable": true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":100000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"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.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.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","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips", "ext.gadget.switcher","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","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=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.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=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <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 property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/5/5c/Standing_waves_on_a_string.gif"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1130"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/5/5c/Standing_waves_on_a_string.gif"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="753"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="603"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Hilbert space - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Hilbert_space"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Hilbert_space&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="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Hilbert_space"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&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_space rootpage-Hilbert_space skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</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="Site"> <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="Main menu" > <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" > Main menu </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">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </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/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </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="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <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="Appearance"> <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="Appearance" > <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" > Appearance </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="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class="">Donate</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=Special:CreateAccount&returnto=Hilbert+space" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</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=Special:UserLogin&returnto=Hilbert+space" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</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="Log in and more options" > <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="Personal tools" > <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" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <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="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Hilbert+space" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Hilbert+space" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition_and_illustration" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_and_illustration"> <div class="vector-toc-text"> 1 Definition and illustration </div> </a> <button aria-controls="toc-Definition_and_illustration-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition and illustration subsection </button> <ul id="toc-Definition_and_illustration-sublist" class="vector-toc-list"> <li id="toc-Motivating_example:_Euclidean_vector_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Motivating_example:_Euclidean_vector_space"> <div class="vector-toc-text"> 1.1 Motivating example: Euclidean vector space </div> </a> <ul id="toc-Motivating_example:_Euclidean_vector_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> 1.2 Definition </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_example:_sequence_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_example:_sequence_spaces"> <div class="vector-toc-text"> 1.3 Second example: sequence spaces </div> </a> <ul id="toc-Second_example:_sequence_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 3 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Lebesgue_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lebesgue_spaces"> <div class="vector-toc-text"> 3.1 Lebesgue spaces </div> </a> <ul id="toc-Lebesgue_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sobolev_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sobolev_spaces"> <div class="vector-toc-text"> 3.2 Sobolev spaces </div> </a> <ul id="toc-Sobolev_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spaces_of_holomorphic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spaces_of_holomorphic_functions"> <div class="vector-toc-text"> 3.3 Spaces of holomorphic functions </div> </a> <ul id="toc-Spaces_of_holomorphic_functions-sublist" class="vector-toc-list"> <li id="toc-Hardy_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hardy_spaces"> <div class="vector-toc-text"> 3.3.1 Hardy spaces </div> </a> <ul id="toc-Hardy_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bergman_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bergman_spaces"> <div class="vector-toc-text"> 3.3.2 Bergman spaces </div> </a> <ul id="toc-Bergman_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 4 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Sturm–Liouville_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sturm–Liouville_theory"> <div class="vector-toc-text"> 4.1 Sturm–Liouville theory </div> </a> <ul id="toc-Sturm–Liouville_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partial_differential_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_differential_equations"> <div class="vector-toc-text"> 4.2 Partial differential equations </div> </a> <ul id="toc-Partial_differential_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ergodic_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ergodic_theory"> <div class="vector-toc-text"> 4.3 Ergodic theory </div> </a> <ul id="toc-Ergodic_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_analysis"> <div class="vector-toc-text"> 4.4 Fourier analysis </div> </a> <ul id="toc-Fourier_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> 4.5 Quantum mechanics </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_theory"> <div class="vector-toc-text"> 4.6 Probability theory </div> </a> <ul id="toc-Probability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Color_perception" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Color_perception"> <div class="vector-toc-text"> 4.7 Color perception </div> </a> <ul id="toc-Color_perception-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 5 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Pythagorean_identity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pythagorean_identity"> <div class="vector-toc-text"> 5.1 Pythagorean identity </div> </a> <ul id="toc-Pythagorean_identity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parallelogram_identity_and_polarization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parallelogram_identity_and_polarization"> <div class="vector-toc-text"> 5.2 Parallelogram identity and polarization </div> </a> <ul id="toc-Parallelogram_identity_and_polarization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Best_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Best_approximation"> <div class="vector-toc-text"> 5.3 Best approximation </div> </a> <ul id="toc-Best_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Duality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Duality"> <div class="vector-toc-text"> 5.4 Duality </div> </a> <ul id="toc-Duality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weakly-convergent_sequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weakly-convergent_sequences"> <div class="vector-toc-text"> 5.5 Weakly-convergent sequences </div> </a> <ul id="toc-Weakly-convergent_sequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Banach_space_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Banach_space_properties"> <div class="vector-toc-text"> 5.6 Banach space properties </div> </a> <ul id="toc-Banach_space_properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Operators_on_Hilbert_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Operators_on_Hilbert_spaces"> <div class="vector-toc-text"> 6 Operators on Hilbert spaces </div> </a> <button aria-controls="toc-Operators_on_Hilbert_spaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Operators on Hilbert spaces subsection </button> <ul id="toc-Operators_on_Hilbert_spaces-sublist" class="vector-toc-list"> <li id="toc-Bounded_operators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bounded_operators"> <div class="vector-toc-text"> 6.1 Bounded operators </div> </a> <ul id="toc-Bounded_operators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unbounded_operators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unbounded_operators"> <div class="vector-toc-text"> 6.2 Unbounded operators </div> </a> <ul id="toc-Unbounded_operators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Constructions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Constructions"> <div class="vector-toc-text"> 7 Constructions </div> </a> <button aria-controls="toc-Constructions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Constructions subsection </button> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Direct_sums" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Direct_sums"> <div class="vector-toc-text"> 7.1 Direct sums </div> </a> <ul id="toc-Direct_sums-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_products" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_products"> <div class="vector-toc-text"> 7.2 Tensor products </div> </a> <ul id="toc-Tensor_products-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orthonormal_bases" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Orthonormal_bases"> <div class="vector-toc-text"> 8 Orthonormal bases </div> </a> <button aria-controls="toc-Orthonormal_bases-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Orthonormal bases subsection </button> <ul id="toc-Orthonormal_bases-sublist" class="vector-toc-list"> <li id="toc-Sequence_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sequence_spaces"> <div class="vector-toc-text"> 8.1 Sequence spaces </div> </a> <ul id="toc-Sequence_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bessel's_inequality_and_Parseval's_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bessel's_inequality_and_Parseval's_formula"> <div class="vector-toc-text"> 8.2 Bessel's inequality and Parseval's formula </div> </a> <ul id="toc-Bessel's_inequality_and_Parseval's_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert_dimension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hilbert_dimension"> <div class="vector-toc-text"> 8.3 Hilbert dimension </div> </a> <ul id="toc-Hilbert_dimension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Separable_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separable_spaces"> <div class="vector-toc-text"> 8.4 Separable spaces </div> </a> <ul id="toc-Separable_spaces-sublist" class="vector-toc-list"> <li id="toc-In_quantum_field_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#In_quantum_field_theory"> <div class="vector-toc-text"> 8.4.1 In quantum field theory </div> </a> <ul id="toc-In_quantum_field_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Orthogonal_complements_and_projections" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Orthogonal_complements_and_projections"> <div class="vector-toc-text"> 9 Orthogonal complements and projections </div> </a> <ul id="toc-Orthogonal_complements_and_projections-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectral_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Spectral_theory"> <div class="vector-toc-text"> 10 Spectral theory </div> </a> <ul id="toc-Spectral_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> 11 In popular culture </div> </a> <ul id="toc-In_popular_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Remarks" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Remarks"> <div class="vector-toc-text"> 13 Remarks </div> </a> <ul id="toc-Remarks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 14 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 15 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 16 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Hilbert space</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 59 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-59" aria-hidden="true" > 59 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/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">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D9%87%D9%8A%D9%84%D8%A8%D8%B1%D8%AA" title="فضاء هيلبرت – Arabic" lang="ar" hreflang="ar" data-title="فضاء هيلبرت" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_de_Hilbert" title="Espaciu de Hilbert – Asturian" lang="ast" hreflang="ast" data-title="Espaciu de Hilbert" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</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 – Azerbaijani" lang="az" hreflang="az" data-title="Hilbert fəzası" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</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">বাংলা</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 – Bashkir" lang="ba" hreflang="ba" data-title="Гильберт арауығы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</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 – Bulgarian" lang="bg" hreflang="bg" data-title="Хилбертово пространство" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</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 – Catalan" lang="ca" hreflang="ca" data-title="Espai de Hilbert" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hilbert%C5%AFv_prostor" title="Hilbertův prostor – Czech" lang="cs" hreflang="cs" data-title="Hilbertův prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hilbertrum" title="Hilbertrum – Danish" lang="da" hreflang="da" data-title="Hilbertrum" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hilbertraum" title="Hilbertraum – German" lang="de" hreflang="de" data-title="Hilbertraum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hilberti_ruum" title="Hilberti ruum – Estonian" lang="et" hreflang="et" data-title="Hilberti ruum" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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 – Greek" lang="el" hreflang="el" data-title="Χώρος Χίλμπερτ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_de_Hilbert" title="Espacio de Hilbert – Spanish" lang="es" hreflang="es" data-title="Espacio de Hilbert" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Hilberta_spaco" title="Hilberta spaco – Esperanto" lang="eo" hreflang="eo" data-title="Hilberta spaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Hilberten_espazio" title="Hilberten espazio – Basque" lang="eu" hreflang="eu" data-title="Hilberten espazio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%87%DB%8C%D9%84%D8%A8%D8%B1%D8%AA" title="فضای هیلبرت – Persian" lang="fa" hreflang="fa" data-title="فضای هیلبرت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_de_Hilbert" title="Espace de Hilbert – French" lang="fr" hreflang="fr" data-title="Espace de Hilbert" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</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 – Galician" lang="gl" hreflang="gl" data-title="Espazo de Hilbert" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9E%90%EB%B2%A0%EB%A5%B4%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="힐베르트 공간 – Korean" lang="ko" hreflang="ko" data-title="힐베르트 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%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 – Armenian" lang="hy" hreflang="hy" data-title="Հիլբերտյան տարածություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_Hilbert" title="Ruang Hilbert – Indonesian" lang="id" hreflang="id" data-title="Ruang Hilbert" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</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 – Icelandic" lang="is" hreflang="is" data-title="Hilbert-rúm" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert – Italian" lang="it" hreflang="it" data-title="Spazio di Hilbert" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%94%D7%99%D7%9C%D7%91%D7%A8%D7%98" title="מרחב הילברט – Hebrew" lang="he" hreflang="he" data-title="מרחב הילברט" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%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 – Kyrgyz" lang="ky" hreflang="ky" data-title="Гильберт мейкиндиги" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</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 – Lithuanian" lang="lt" hreflang="lt" data-title="Hilberto erdvė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hilbert-t%C3%A9r" title="Hilbert-tér – Hungarian" lang="hu" hreflang="hu" data-title="Hilbert-tér" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_Hilbert" title="Ruang Hilbert – Malay" lang="ms" hreflang="ms" data-title="Ruang Hilbert" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</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 – Mongolian" lang="mn" hreflang="mn" data-title="Хилбертийн орон зай" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hilbertruimte" title="Hilbertruimte – Dutch" lang="nl" hreflang="nl" data-title="Hilbertruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%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 – Japanese" lang="ja" hreflang="ja" data-title="ヒルベルト空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hilbert-rom" title="Hilbert-rom – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hilbert-rom" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hilbertrom" title="Hilbertrom – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Hilbertrom" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gilbert_fazosi" title="Gilbert fazosi – Uzbek" lang="uz" hreflang="uz" data-title="Gilbert fazosi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</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 – Punjabi" lang="pa" hreflang="pa" data-title="ਹਿਲਬਰਟ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%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">پنجابی</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 – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń Hilberta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_de_Hilbert" title="Espaço de Hilbert – Portuguese" lang="pt" hreflang="pt" data-title="Espaço de Hilbert" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</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 – Romanian" lang="ro" hreflang="ro" data-title="Spațiu Hilbert" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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 – Russian" lang="ru" hreflang="ru" data-title="Гильбертово пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Hilbert_space" title="Hilbert space – Scots" lang="sco" hreflang="sco" data-title="Hilbert space" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</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 – Albanian" lang="sq" hreflang="sq" data-title="Hapësira e Hilbertit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</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">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hilbertov_priestor" title="Hilbertov priestor – Slovak" lang="sk" hreflang="sk" data-title="Hilbertov priestor" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hilbertov_prostor" title="Hilbertov prostor – Slovenian" lang="sl" hreflang="sl" data-title="Hilbertov prostor" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%DA%BE%DB%8C%D9%84%D8%A8%DB%8E%D8%B1%D8%AA" title="بۆشاییی ھیلبێرت – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بۆشاییی ھیلبێرت" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%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 – Serbian" lang="sr" hreflang="sr" data-title="Хилбертов простор" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Hilbertov_prostor" title="Hilbertov prostor – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Hilbertov prostor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hilbertin_avaruus" title="Hilbertin avaruus – Finnish" lang="fi" hreflang="fi" data-title="Hilbertin avaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hilbertrum" title="Hilbertrum – Swedish" lang="sv" hreflang="sv" data-title="Hilbertrum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espasyong_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">Tagalog</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 – Turkish" lang="tr" hreflang="tr" data-title="Hilbert uzayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%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 – Ukrainian" lang="uk" hreflang="uk" data-title="Гільбертів простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</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 – Vietnamese" lang="vi" hreflang="vi" data-title="Không gian Hilbert" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%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">文言</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" lang="wuu" hreflang="wuu" data-title="希尔伯特空间" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9B%82%E6%8B%94%E7%A9%BA%E9%96%93" title="囂拔空間 – Cantonese" lang="yue" hreflang="yue" data-title="囂拔空間" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4" title="希尔伯特空间 – Chinese" lang="zh" hreflang="zh" data-title="希尔伯特空间" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q190056#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Hilbert_space" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Hilbert_space" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</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="Change language variant" > <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" >English </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="Views"> <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_space">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hilbert_space&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hilbert_space&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <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="Tools" > <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" >Tools </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">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </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_space">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hilbert_space&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hilbert_space&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Hilbert_space" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Hilbert_space" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Hilbert_space&oldid=1255844830" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Hilbert_space&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Hilbert_space&id=1255844830&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHilbert_space">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FHilbert_space">Download QR code</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"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Hilbert_space&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Hilbert_space&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </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">Wikimedia Commons</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="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</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="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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-good-star" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Good_articles*" title="This is a good article. Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of topological vector space</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the space-filling curve, see <a href="/wiki/Hilbert_curve" title="Hilbert curve">Hilbert curve</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Standing_waves_on_a_string.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Standing_waves_on_a_string.gif/220px-Standing_waves_on_a_string.gif" decoding="async" width="220" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Standing_waves_on_a_string.gif/330px-Standing_waves_on_a_string.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/5/5c/Standing_waves_on_a_string.gif 2x" data-file-width="360" data-file-height="339" /></a><figcaption>The state of a <a href="/wiki/Vibrating_string" class="mw-redirect" title="Vibrating string">vibrating string</a> can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct <a href="/wiki/Overtone" title="Overtone">overtones</a> is given by the projection of the point onto the <a href="/wiki/Coordinate_axes" class="mw-redirect" title="Coordinate axes">coordinate axes</a> in the space.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, Hilbert spaces (named after <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>) allow the methods of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and <a href="/wiki/Calculus" title="Calculus">calculus</a> to be generalized from (finite-dimensional) <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector spaces</a> to spaces that may be <a href="/wiki/Infinite-dimensional" class="mw-redirect" title="Infinite-dimensional">infinite-dimensional</a>. Hilbert spaces arise naturally and frequently in mathematics and <a href="/wiki/Physics" title="Physics">physics</a>, typically as <a href="/wiki/Function_space" title="Function space">function spaces</a>. Formally, a Hilbert space is a <a href="/wiki/Vector_space" title="Vector space">vector space</a> equipped with an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> that induces a <a href="/wiki/Distance_function" class="mw-redirect" title="Distance function">distance function</a> for which the space is a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>. A Hilbert space is a special case of a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. Hilbert spaces were studied beginning in the first decade of the 20th century by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, <a href="/wiki/Erhard_Schmidt" title="Erhard Schmidt">Erhard Schmidt</a>, and <a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Frigyes Riesz</a>. They are indispensable tools in the theories of <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>, <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">quantum mechanics</a>, <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> (which includes applications to <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> and <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>), and <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a> (which forms the mathematical underpinning of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>). <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include <a href="/wiki/Square-integrable_function" title="Square-integrable function">spaces of square-integrable functions</a>, <a href="/wiki/Sequence_space" title="Sequence space">spaces of sequences</a>, <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a> consisting of <a href="/wiki/Generalized_function" title="Generalized function">generalized functions</a>, and <a href="/wiki/Hardy_space" title="Hardy space">Hardy spaces</a> of <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a>. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> and <a href="/wiki/Parallelogram_law" title="Parallelogram law">parallelogram law</a> hold in a Hilbert space. At a deeper level, <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection</a> onto a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> plays a significant role in <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">optimization</a> problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>, in analogy with <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> in classical geometry. When this <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> is <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a>, it allows identifying the Hilbert space with the space of the <a href="/wiki/Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequences</a> that are <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-summable</a>. The latter space is often in the older literature referred to as the Hilbert space. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_illustration">Definition and illustration</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=1" title="Edit section: Definition and illustration">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Motivating_example:_Euclidean_vector_space">Motivating example: Euclidean vector space</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=2" title="Edit section: Motivating example: Euclidean vector space">edit</a>]</div> One of the most familiar examples of a Hilbert space is the <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector space</a> consisting of three-dimensional <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a>, denoted by R3, and equipped with the <a href="/wiki/Dot_product" title="Dot product">dot product</a>. The dot product takes two vectors x and y, and produces a real number x ⋅ y. If x and y are represented in <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, then the dot product is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6c73091bcfb40e8ae18ba63c544d4bd4d690b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:39.439ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.}"> The dot product satisfies the properties<a href="#cite_note-1">[1]</a> <ol><li>It is <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric</a> in x and y: x ⋅ y = y ⋅ x.</li> <li>It is <a href="/wiki/Linear_function" title="Linear function">linear</a> in its first argument: (ax1 + bx2) ⋅ y = a(x1 ⋅ y) + b(x2 ⋅ y) for any <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a> a, b, and vectors x1, x2, and y.</li> <li>It is <a href="/wiki/Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">positive definite</a>: for all vectors x, x ⋅ x ≥ 0 , with equality <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> x = 0.</li></ol> An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a>. A <a href="/wiki/Vector_space" title="Vector space">vector space</a> equipped with such an inner product is known as a (real) <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>. Every finite-dimensional inner product space is also a Hilbert space.<a href="#cite_note-2">[2]</a> The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>) of a vector, denoted ‖x‖, and to the angle θ between two vectors x and y by means of the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff5157fde2381e9dcd0a006d3e1d0575512afbd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.855ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.}"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Completeness_in_Hilbert_space.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Completeness_in_Hilbert_space.png/220px-Completeness_in_Hilbert_space.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Completeness_in_Hilbert_space.png/330px-Completeness_in_Hilbert_space.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Completeness_in_Hilbert_space.png/440px-Completeness_in_Hilbert_space.png 2x" data-file-width="742" data-file-height="459" /></a><figcaption>Completeness means that a series of vectors (in blue) results in a <a href="/wiki/Well_defined" class="mw-redirect" title="Well defined">well-defined</a> net displacement vector (in orange).</figcaption></figure> <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a> in Euclidean space relies on the ability to compute <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a>, and to have useful criteria for concluding that limits exist. A <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">mathematical series</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3d791a8b023df5a980200e276eec54d69eb2b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.372ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}}"> consisting of vectors in R3 is <a href="/wiki/Absolute_convergence" title="Absolute convergence">absolutely convergent</a> provided that the sum of the lengths converges as an ordinary series of real numbers:<a href="#cite_note-3">[3]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c49858bb1785370ae3843e6ea5398f612524f806" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.023ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.}"> Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Biggl \|}\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}{\Biggr \|}\to 0\quad {\text{as }}N\to \infty \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo symmetric="true" maxsize="2.470em" minsize="2.470em">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo symmetric="true" maxsize="2.470em" minsize="2.470em">‖</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Biggl \|}\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}{\Biggr \|}\to 0\quad {\text{as }}N\to \infty \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26b075a3dcc6d237c8c141a155f076338daaaed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.809ex; height:7.509ex;" alt="{\displaystyle {\Biggl \|}\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}{\Biggr \|}\to 0\quad {\text{as }}N\to \infty \,.}"> This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. The <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> denoted by C is equipped with a notion of magnitude, the <a href="/wiki/Absolute_value" title="Absolute value">complex modulus</a> |z|, which is defined as the square root of the product of z with its <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|^{2}=z{\overline {z}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|^{2}=z{\overline {z}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f9ec3c8dc777c161389ccaf3e175e8993b4a19" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.864ex; height:3.343ex;" alt="{\displaystyle |z|^{2}=z{\overline {z}}\,.}"> If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21888cd022a8dfd5d07d8257d0fd25e2823b5bce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.277ex; height:4.843ex;" alt="{\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.}"> The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle z,w\rangle =z{\overline {w}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle z,w\rangle =z{\overline {w}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c95ae0bc59457f9b44a262586ab7caa2542d11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.595ex; height:2.843ex;" alt="{\displaystyle \langle z,w\rangle =z{\overline {w}}\,.}"> This is complex-valued. The real part of ⟨z, w⟩ gives the usual two-dimensional Euclidean <a href="/wiki/Dot_product" title="Dot product">dot product</a>. A second example is the space C2 whose elements are pairs of complex numbers z = (z1, z2). Then an inner product of z with another such vector w = (w1, w2) is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2f682a0dac408d4cd1e636a033e39c5021d757" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.506ex; height:2.843ex;" alt="{\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.}"> The real part of ⟨z, w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>w</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f310d466f226a09af8f8593ff396e1f0401251" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.438ex; height:3.676ex;" alt="{\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.}"> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=3" title="Edit section: Definition">edit</a>]</div> A Hilbert space is a <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a> that is also a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a> with respect to the distance function <a href="/wiki/Induced_topology" class="mw-redirect" title="Induced topology">induced</a> by the inner product.<a href="#cite_note-General-4">[4]</a> To say that a complex vector space H is a complex inner product space means that there is an inner product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"> associating a complex number to each pair of elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,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,y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"> of H that satisfies the following properties: <ol><li>The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the inner product of the swapped elements: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb947b57df47d548a0ff388d1af8fb489e90f3ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.904ex; height:3.676ex;" alt="{\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}\,.}"> Importantly, this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f081a4822ec7bdff2661ee2070b70a3406cf70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle }"> is a real number.</li> <li>The inner product is <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear</a> in its first<a href="#cite_note-5">[nb 1]</a> argument. For all complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>a</mi> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>b</mi> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c1164bfad1024b720cd7262f99f77063676ab12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.8ex; height:2.843ex;" alt="{\displaystyle \langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle \,.}"></li> <li>The inner product of an element with itself is <a href="/wiki/Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">positive definite</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{4}\langle x,x\rangle >0&\quad {\text{ if }}x\neq 0,\\\langle x,x\rangle =0&\quad {\text{ if }}x=0\,.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>></mo> <mn>0</mn> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{4}\langle x,x\rangle >0&\quad {\text{ if }}x\neq 0,\\\langle x,x\rangle =0&\quad {\text{ if }}x=0\,.\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04e01914ed2ed909cbfaaa2dafd141857e258e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.982ex; height:6.176ex;" alt="{\displaystyle {\begin{alignedat}{4}\langle x,x\rangle >0&\quad {\text{ if }}x\neq 0,\\\langle x,x\rangle =0&\quad {\text{ if }}x=0\,.\end{alignedat}}}"></li></ol> It follows from properties 1 and 2 that a complex inner product is <a href="/wiki/Antilinear_map" title="Antilinear map">antilinear</a>, also called conjugate linear, in its second argument, meaning that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>a</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648a87771f6ed2eae5f1a7867493312b6292ba71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.725ex; height:3.009ex;" alt="{\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.}"> A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear map</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (H,H,\langle \cdot ,\cdot \rangle )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>H</mi> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (H,H,\langle \cdot ,\cdot \rangle )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bee7511626db6a54e1af12d29ff989d2aff8b7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.141ex; height:2.843ex;" alt="{\displaystyle (H,H,\langle \cdot ,\cdot \rangle )}"> will form a <a href="/wiki/Dual_system" title="Dual system">dual system</a>.<a href="#cite_note-FOOTNOTESchaeferWolff1999122–202-6">[5]</a> The <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> is the real-valued function <math display="block" 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> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d803bc001c215a1be59acf43e6606b1d774ea8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.613ex; height:4.843ex;" alt="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}\,,}"> and the distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> between two points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,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,y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"> in H is defined in terms of the norm by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4401717fb6ad3702ce9b40126a159d8756a8fed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.244ex; height:4.843ex;" alt="{\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.}"> That this function is a distance function means firstly that it is symmetric in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bd9829c9ef4adcb0f9f5d53b27463a873a8e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y,}"> secondly that the distance between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and itself is zero, and otherwise the distance between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> must be positive, and lastly that the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef67169190c37bb6b853a1304d1c5ead6bad92b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.297ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.}"> <dl><dd><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/File:Triangle_inequality_in_a_metric_space.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Triangle_inequality_in_a_metric_space.svg/300px-Triangle_inequality_in_a_metric_space.svg.png" decoding="async" width="300" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Triangle_inequality_in_a_metric_space.svg/450px-Triangle_inequality_in_a_metric_space.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Triangle_inequality_in_a_metric_space.svg/600px-Triangle_inequality_in_a_metric_space.svg.png 2x" data-file-width="456" data-file-height="213" /></a><figcaption></figcaption></figure></dd></dl> This last property is ultimately a consequence of the more fundamental <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>, which asserts <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea56af9f54f8732fcbe7204ffdcebaeb16a4ba8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.856ex; height:2.843ex;" alt="{\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|}"> with equality if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> are <a href="/wiki/Linear_independence" title="Linear independence">linearly dependent</a>. With a distance function defined in this way, any inner product space is a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, and sometimes is known as a pre-Hilbert space.<a href="#cite_note-7">[6]</a> Any pre-Hilbert space that is additionally also a <a href="/wiki/Complete_space" class="mw-redirect" title="Complete space">complete space</a> is a Hilbert space.<a href="#cite_note-8">[7]</a> The <a href="/wiki/Complete_metric_space" title="Complete metric space">completeness</a> of H is expressed using a form of the <a href="/wiki/Cauchy_criterion" class="mw-redirect" title="Cauchy criterion">Cauchy criterion</a> for sequences in H: a pre-Hilbert space H is complete if every <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">converges with respect to this norm</a> to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }u_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }u_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eab948361735efa594296bd04d873a8485149c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.16ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }u_{k}}"> <a href="/wiki/Absolute_convergence" title="Absolute convergence">converges absolutely</a> in the sense that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a193b4cda60532c04dc0900d7d30353145ffcab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.942ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,}"> then the series converges in H, in the sense that the <a href="/wiki/Partial_sums" class="mw-redirect" title="Partial sums">partial sums</a> converge to an element of H.<a href="#cite_note-9">[8]</a> As a complete normed space, Hilbert spaces are by definition also <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>. As such they are <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a>, in which <a href="/wiki/Topology" title="Topology">topological</a> notions like the <a href="/wiki/Open_set" title="Open set">openness</a> and <a href="/wiki/Closed_set" title="Closed set">closedness</a> of subsets are <a href="/wiki/Well-defined_expression" title="Well-defined expression">well defined</a>. Of special importance is the notion of a closed <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of a Hilbert space that, with the inner product induced by <a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">restriction</a>, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right. <div class="mw-heading mw-heading3"><h3 id="Second_example:_sequence_spaces">Second example: sequence spaces</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=4" title="Edit section: Second example: sequence spaces">edit</a>]</div> The <a href="/wiki/Sequence_space" title="Sequence space">sequence space</a> l2 consists of all <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">infinite sequences</a> z = (z1, z2, …) of complex numbers such that the following series <a href="/wiki/Convergent_series" title="Convergent series">converges</a>:<a href="#cite_note-Stein_2005-10">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e56a3c8d984a07aea65427989643bcd774fc8ab0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.39ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}}"> The inner product on l2 is defined by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <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 mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e4a63265ba175624e53a0dca1caa094d62114e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.134ex; height:6.843ex;" alt="{\displaystyle \langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,}"> This second series converges as a consequence of the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a> and the convergence of the previous series. Completeness of the space holds provided that whenever a series of elements from l2 converges absolutely (in norm), then it converges to an element of l2. The proof is basic in <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).<a href="#cite_note-11">[10]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=5" title="Edit section: History">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hilbert.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Hilbert.jpg/220px-Hilbert.jpg" decoding="async" width="220" height="298" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Hilbert.jpg/330px-Hilbert.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/79/Hilbert.jpg 2x" data-file-width="437" data-file-height="592" /></a><figcaption><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a></figcaption></figure> Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to <a href="/wiki/Mathematician" title="Mathematician">mathematicians</a> and <a href="/wiki/Physicist" title="Physicist">physicists</a>. In particular, the idea of an <a href="/wiki/Vector_space" title="Vector space">abstract linear space (vector space)</a> had gained some traction towards the end of the 19th century:<a href="#cite_note-12">[11]</a> this is a space whose elements can be added together and multiplied by scalars (such as <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a> or <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>) without necessarily identifying these elements with <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">"geometric" vectors</a>, such as position and momentum vectors in physical systems. Other objects studied by mathematicians at the turn of the 20th century, in particular spaces of <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequences</a> (including <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>) and spaces of functions,<a href="#cite_note-13">[12]</a> can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors. In the first decade of the 20th century, parallel developments led to the introduction of Hilbert spaces. The first of these was the observation, which arose during <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Erhard_Schmidt" title="Erhard Schmidt">Erhard Schmidt</a>'s study of <a href="/wiki/Integral_equations" class="mw-redirect" title="Integral equations">integral equations</a>,<a href="#cite_note-14">[13]</a> that two <a href="/wiki/Square-integrable" class="mw-redirect" title="Square-integrable">square-integrable</a> real-valued functions f and g on an interval [a, b] have an inner product <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(x)g(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(x)g(x)\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3b8c1935031c7769ee297cf76cba993af6de5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.807ex; height:6.343ex;" alt="{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(x)g(x)\,\mathrm {d} x}"></dd></dl> which has many of the familiar properties of the Euclidean dot product. In particular, the idea of an <a href="/wiki/Orthogonality" title="Orthogonality">orthogonal</a> family of functions has meaning. Schmidt exploited the similarity of this inner product with the usual dot product to prove an analog of the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral decomposition</a> for an operator of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\mapsto \int _{a}^{b}K(x,y)f(y)\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\mapsto \int _{a}^{b}K(x,y)f(y)\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dee2bd40943738c454875f27b51c039baf5ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.293ex; height:6.343ex;" alt="{\displaystyle f(x)\mapsto \int _{a}^{b}K(x,y)f(y)\,\mathrm {d} y}"></dd></dl> where K is a continuous function symmetric in x and y. The resulting <a href="/wiki/Eigenfunction_expansion" class="mw-redirect" title="Eigenfunction expansion">eigenfunction expansion</a> expresses the function K as a series of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x,y)=\sum _{n}\lambda _{n}\varphi _{n}(x)\varphi _{n}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x,y)=\sum _{n}\lambda _{n}\varphi _{n}(x)\varphi _{n}(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921e2db751333a14035d636bcf05c6a167344ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.39ex; height:5.509ex;" alt="{\displaystyle K(x,y)=\sum _{n}\lambda _{n}\varphi _{n}(x)\varphi _{n}(y)}"></dd></dl> where the functions φn are orthogonal in the sense that ⟨φn, φm⟩ = 0 for all n ≠ m. The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness.<a href="#cite_note-15">[14]</a> The second development was the <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">Lebesgue integral</a>, an alternative to the <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a> introduced by <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a> in 1904.<a href="#cite_note-16">[15]</a> The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, <a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Frigyes Riesz</a> and <a href="/wiki/Ernst_Sigismund_Fischer" title="Ernst Sigismund Fischer">Ernst Sigismund Fischer</a> independently proved that the space L2 of square Lebesgue-integrable functions is a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>.<a href="#cite_note-17">[16]</a> As a consequence of the interplay between geometry and completeness, the 19th century results of <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a>, <a href="/wiki/Friedrich_Bessel" class="mw-redirect" title="Friedrich Bessel">Friedrich Bessel</a> and <a href="/wiki/Marc-Antoine_Parseval" title="Marc-Antoine Parseval">Marc-Antoine Parseval</a> on <a href="/wiki/Trigonometric_series" title="Trigonometric series">trigonometric series</a> easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the <a href="/wiki/Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a>.<a href="#cite_note-18">[17]</a> Further basic results were proved in the early 20th century. For example, the <a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a> was independently established by <a href="/wiki/Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Maurice Fréchet</a> and <a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Frigyes Riesz</a> in 1907.<a href="#cite_note-19">[18]</a> <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> coined the term abstract Hilbert space in his work on unbounded <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Hermitian operators</a>.<a href="#cite_note-20">[19]</a> Although other mathematicians such as <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> and <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them.<a href="#cite_note-21">[20]</a> Von Neumann later used them in his seminal work on the foundations of quantum mechanics,<a href="#cite_note-22">[21]</a> and in his continued work with <a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a>. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.<a href="#cite_note-Weyl31-23">[22]</a> The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">mathematical formulations of quantum mechanics</a>.<a href="#cite_note-24">[23]</a> In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are <a href="/wiki/Hermitian_operator" class="mw-redirect" title="Hermitian operator">hermitian operators</a> on that space, the <a href="/wiki/Symmetry" title="Symmetry">symmetries</a> of the system are <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operators</a>, and <a href="/wiki/Quantum_measurement" class="mw-redirect" title="Quantum measurement">measurements</a> are <a href="/wiki/Orthogonal_projection" class="mw-redirect" title="Orthogonal projection">orthogonal projections</a>. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the <a href="/wiki/Unitary_representation" title="Unitary representation">unitary</a> <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> of <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, initiated in the 1928 work of Hermann Weyl.<a href="#cite_note-Weyl31-23">[22]</a> On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space (<a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann classical mechanics</a>) and that certain properties of classical <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a> can be analyzed using Hilbert space techniques in the framework of <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a>.<a href="#cite_note-von_Neumann_1932-25">[24]</a> The algebra of <a href="/wiki/Observable" title="Observable">observables</a> in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>'s <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> formulation of quantum theory.<a href="#cite_note-26">[25]</a> Von Neumann began investigating <a href="/wiki/Operator_algebra" title="Operator algebra">operator algebras</a> in the 1930s, as <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as <a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">von Neumann algebras</a>.<a href="#cite_note-27">[26]</a> In the 1940s, <a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Israel Gelfand</a>, <a href="/wiki/Mark_Naimark" title="Mark Naimark">Mark Naimark</a> and <a href="/wiki/Irving_Segal" title="Irving Segal">Irving Segal</a> gave a definition of a kind of operator algebras called <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a> that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras.<a href="#cite_note-28">[27]</a> These techniques are now basic in abstract harmonic analysis and representation theory. <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=6" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Lebesgue_spaces">Lebesgue spaces</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=7" title="Edit section: Lebesgue spaces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lp_space" title="Lp space">Lp space</a></div> Lebesgue spaces are <a href="/wiki/Function_space" title="Function space">function spaces</a> associated to <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure spaces</a> (X, M, μ), where X is a set, M is a <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">σ-algebra</a> of subsets of X, and μ is a <a href="/wiki/Countably_additive_measure" class="mw-redirect" title="Countably additive measure">countably additive measure</a> on M. Let L2(X, μ) be the space of those complex-valued measurable functions on X for which the <a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integral</a> of the square of the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the function is finite, i.e., for a function f in L2(X, μ), <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee85db694c36e91a5e3cb04a30106bc3309da88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.089ex; height:5.676ex;" alt="{\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,}"> and where functions are identified if and only if they differ only on a <a href="/wiki/Null_set" title="Null set">set of measure zero</a>. The inner product of functions f and g in L2(X, μ) is then defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444efb40cac89ff9e517f21cabefac4b4c461817" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.186ex; height:5.676ex;" alt="{\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)}"> or <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <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> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759d01e4c6dcc1d76d34c5bf653cf220818a464d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.22ex; height:5.676ex;" alt="{\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,}"> where the second form (conjugation of the first element) is commonly found in the <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> literature. For f and g in L2, the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L2 is in fact complete.<a href="#cite_note-29">[28]</a> The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integrable</a>.<a href="#cite_note-30">[29]</a> The Lebesgue spaces appear in many natural settings. The spaces L2(R) and L2([0,1]) of square-integrable functions with respect to the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if w is any positive measurable function, the space of all measurable functions f on the interval [0, 1] satisfying <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>w</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06662023c777cfa7c3cda7e5fa8e2e3e390f053f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.435ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty }"> is called the <a href="/wiki/Lp_space#Weighted_Lp_spaces" title="Lp space">weighted L2 space</a> L2 w([0, 1]), and w is called the weight function. The inner product is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mi>w</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3305f5510dd0b3311769c19e47bc8f2b0c4e832d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.915ex; height:6.176ex;" alt="{\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.}"> The weighted space L2 w([0, 1]) is identical with the Hilbert space L2([0, 1], μ) where the measure μ of a Lebesgue-measurable set A is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/536c89546b2b4876f1c6835d87700c09fd984100" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.063ex; height:5.676ex;" alt="{\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.}"> Weighted L2 spaces like this are frequently used to study <a href="/wiki/Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a>, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.<a href="#cite_note-31">[30]</a> <div class="mw-heading mw-heading3"><h3 id="Sobolev_spaces">Sobolev spaces</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=8" title="Edit section: Sobolev spaces">edit</a>]</div> <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a>, denoted by Hs or Ws, 2, are Hilbert spaces. These are a special kind of <a href="/wiki/Function_space" title="Function space">function space</a> in which <a href="/wiki/Derivative" title="Derivative">differentiation</a> may be performed, but that (unlike other <a href="/wiki/Banach_spaces" class="mw-redirect" title="Banach spaces">Banach spaces</a> such as the <a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder spaces</a>) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>.<a href="#cite_note-BeJoSc81-32">[31]</a> They also form the basis of the theory of <a href="/wiki/Direct_method_in_calculus_of_variations" class="mw-redirect" title="Direct method in calculus of variations">direct methods in the calculus of variations</a>.<a href="#cite_note-33">[32]</a> For s a non-negative <a href="/wiki/Integer" title="Integer">integer</a> and Ω ⊂ Rn, the Sobolev space Hs(Ω) contains L2 functions whose <a href="/wiki/Weak_derivative" title="Weak derivative">weak derivatives</a> of order up to s are also L2. The inner product in Hs(Ω) is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>D</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50aeb4e11264166dd7a52c8a92ff01274b07ced5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:77.33ex; height:5.676ex;" alt="{\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x}"> where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when s is not an integer. Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If Ω is a suitable domain, then one can define the Sobolev space Hs(Ω) as the space of <a href="/wiki/Bessel_potential" title="Bessel potential">Bessel potentials</a>;<a href="#cite_note-34">[33]</a> roughly, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{s}(\Omega )=\left\{(1-\Delta )^{-s/2}f\mathrel {\Big |} f\in L^{2}(\Omega )\right\}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>f</mi> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{s}(\Omega )=\left\{(1-\Delta )^{-s/2}f\mathrel {\Big |} f\in L^{2}(\Omega )\right\}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d29c7b84760df017ba43e6ddfaf97c373b67fc92" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.348ex; height:4.843ex;" alt="{\displaystyle H^{s}(\Omega )=\left\{(1-\Delta )^{-s/2}f\mathrel {\Big |} f\in L^{2}(\Omega )\right\}\,.}"> Here Δ is the Laplacian and (1 − Δ)−s / 2 is understood in terms of the <a href="/wiki/Spectral_mapping_theorem" class="mw-redirect" title="Spectral mapping theorem">spectral mapping theorem</a>. Apart from providing a workable definition of Sobolev spaces for non-integer s, this definition also has particularly desirable properties under the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> that make it ideal for the study of <a href="/wiki/Pseudodifferential_operator" class="mw-redirect" title="Pseudodifferential operator">pseudodifferential operators</a>. Using these methods on a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, one can obtain for instance the <a href="/wiki/Hodge_decomposition" class="mw-redirect" title="Hodge decomposition">Hodge decomposition</a>, which is the basis of <a href="/wiki/Hodge_theory" title="Hodge theory">Hodge theory</a>.<a href="#cite_note-35">[34]</a> <div class="mw-heading mw-heading3"><h3 id="Spaces_of_holomorphic_functions">Spaces of holomorphic functions</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=9" title="Edit section: Spaces of holomorphic functions">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Hardy_spaces">Hardy spaces</h4>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=10" title="Edit section: Hardy spaces">edit</a>]</div> The <a href="/wiki/Hardy_space" title="Hardy space">Hardy spaces</a> are function spaces, arising in <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> and <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>, whose elements are certain <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> in a complex domain.<a href="#cite_note-36">[35]</a> Let U denote the <a href="/wiki/Unit_disc" class="mw-redirect" title="Unit disc">unit disc</a> in the complex plane. Then the Hardy space H2(U) is defined as the space of holomorphic functions f on U such that the means <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{r}(f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f{\bigl (}re^{i\theta }{\bigr )}\right|^{2}\,\mathrm {d} \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{r}(f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f{\bigl (}re^{i\theta }{\bigr )}\right|^{2}\,\mathrm {d} \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d48a7b8ea6e2c4f63b249ef1921e472f615354" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.209ex; height:6.176ex;" alt="{\displaystyle M_{r}(f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f{\bigl (}re^{i\theta }{\bigr )}\right|^{2}\,\mathrm {d} \theta }"> remain bounded for r < 1. The norm on this Hardy space is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|f\right\|_{2}=\lim _{r\to 1}{\sqrt {M_{r}(f)}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo symmetric="true">‖</mo> <mi>f</mi> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|f\right\|_{2}=\lim _{r\to 1}{\sqrt {M_{r}(f)}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246ff4232c927287a6ed5b599c9fa846a1529085" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.046ex; height:5.009ex;" alt="{\displaystyle \left\|f\right\|_{2}=\lim _{r\to 1}{\sqrt {M_{r}(f)}}\,.}"> Hardy spaces in the disc are related to Fourier series. A function f is in H2(U) if and only if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2163356e45f7c36725ccdd94146d0f9a204b4fad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.774ex; height:6.843ex;" alt="{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0205b868913347fe8e885d451d575382c60941d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.994ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \,.}"> Thus H2(U) consists of those functions that are L2 on the circle, and whose negative frequency Fourier coefficients vanish. <div class="mw-heading mw-heading4"><h4 id="Bergman_spaces">Bergman spaces</h4>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=11" title="Edit section: Bergman spaces">edit</a>]</div> The <a href="/wiki/Bergman_space" title="Bergman space">Bergman spaces</a> are another family of Hilbert spaces of holomorphic functions.<a href="#cite_note-37">[36]</a> Let D be a bounded open set in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> (or a higher-dimensional complex space) and let L2, h(D) be the space of holomorphic functions f in D that are also in L2(D) in the sense that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|^{2}=\int _{D}|f(z)|^{2}\,\mathrm {d} \mu (z)<\infty \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|^{2}=\int _{D}|f(z)|^{2}\,\mathrm {d} \mu (z)<\infty \,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abfa454eccbd44a2f8b837454134d9770b38577" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.988ex; height:5.676ex;" alt="{\displaystyle \|f\|^{2}=\int _{D}|f(z)|^{2}\,\mathrm {d} \mu (z)<\infty \,,}"> where the integral is taken with respect to the Lebesgue measure in D. Clearly L2, h(D) is a subspace of L2(D); in fact, it is a <a href="/wiki/Closed_set" title="Closed set">closed</a> subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on <a href="/wiki/Compact_space" title="Compact space">compact</a> subsets K of D, that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{z\in K}\left|f(z)\right|\leq C_{K}\left\|f\right\|_{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈</mo> <mi>K</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <msub> <mrow> <mo symmetric="true">‖</mo> <mi>f</mi> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{z\in K}\left|f(z)\right|\leq C_{K}\left\|f\right\|_{2}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7243a8cb764807f9a5d13eb03ada405869f8642d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.503ex; height:4.509ex;" alt="{\displaystyle \sup _{z\in K}\left|f(z)\right|\leq C_{K}\left\|f\right\|_{2}\,,}"> which in turn follows from <a href="/wiki/Cauchy%27s_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a>. Thus convergence of a sequence of holomorphic functions in L2(D) implies also <a href="/wiki/Compact_convergence" title="Compact convergence">compact convergence</a>, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function f at a point of D is actually continuous on L2, h(D). The Riesz representation theorem implies that the evaluation functional can be represented as an element of L2, h(D). Thus, for every z ∈ D, there is a function ηz ∈ L2, h(D) such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\int _{D}f(\zeta ){\overline {\eta _{z}(\zeta )}}\,\mathrm {d} \mu (\zeta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=\int _{D}f(\zeta ){\overline {\eta _{z}(\zeta )}}\,\mathrm {d} \mu (\zeta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ad03af32f07660d72581836b127864b66754ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.892ex; height:5.676ex;" alt="{\displaystyle f(z)=\int _{D}f(\zeta ){\overline {\eta _{z}(\zeta )}}\,\mathrm {d} \mu (\zeta )}"> for all f ∈ L2, h(D). The integrand <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\zeta ,z)={\overline {\eta _{z}(\zeta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(\zeta ,z)={\overline {\eta _{z}(\zeta )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b706e60c2651a125a998c77e079cda951306c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.367ex; height:3.676ex;" alt="{\displaystyle K(\zeta ,z)={\overline {\eta _{z}(\zeta )}}}"> is known as the <a href="/wiki/Bergman_kernel" title="Bergman kernel">Bergman kernel</a> of D. This <a href="/wiki/Integral_kernel" class="mw-redirect" title="Integral kernel">integral kernel</a> satisfies a reproducing property <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\int _{D}f(\zeta )K(\zeta ,z)\,\mathrm {d} \mu (\zeta )\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(z)=\int _{D}f(\zeta )K(\zeta ,z)\,\mathrm {d} \mu (\zeta )\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/537dd2cecfc6886382c7172984995ebeb0fe6be1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.842ex; height:5.676ex;" alt="{\displaystyle f(z)=\int _{D}f(\zeta )K(\zeta ,z)\,\mathrm {d} \mu (\zeta )\,.}"> A Bergman space is an example of a <a href="/wiki/Reproducing_kernel_Hilbert_space" title="Reproducing kernel Hilbert space">reproducing kernel Hilbert space</a>, which is a Hilbert space of functions along with a kernel K(ζ, z) that verifies a reproducing property analogous to this one. The Hardy space H2(D) also admits a reproducing kernel, known as the <a href="/wiki/Szeg%C5%91_kernel" title="Szegő kernel">Szegő kernel</a>.<a href="#cite_note-38">[37]</a> Reproducing kernels are common in other areas of mathematics as well. For instance, in <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> the <a href="/wiki/Poisson_kernel" title="Poisson kernel">Poisson kernel</a> is a reproducing kernel for the Hilbert space of square-integrable <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic functions</a> in the <a href="/wiki/Unit_ball" class="mw-redirect" title="Unit ball">unit ball</a>. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=12" title="Edit section: Applications">edit</a>]</div> Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like <a href="/wiki/Projection_operator" class="mw-redirect" title="Projection operator">projection</a> and <a href="/wiki/Change_of_basis" title="Change of basis">change of basis</a> from their usual finite dimensional setting. In particular, the <a href="/wiki/Spectral_theory" title="Spectral theory">spectral theory</a> of <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint</a> <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> on a Hilbert space generalizes the usual <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">spectral decomposition</a> of a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>, and this often plays a major role in applications of the theory to other areas of mathematics and physics. <div class="mw-heading mw-heading3"><h3 id="Sturm–Liouville_theory">Sturm–Liouville theory</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=13" title="Edit section: Sturm–Liouville theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a> and <a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Harmonic_partials_on_strings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/220px-Harmonic_partials_on_strings.svg.png" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/330px-Harmonic_partials_on_strings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Harmonic_partials_on_strings.svg/440px-Harmonic_partials_on_strings.svg.png 2x" data-file-width="620" data-file-height="590" /></a><figcaption>The <a href="/wiki/Overtone" title="Overtone">overtones</a> of a vibrating string. These are <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of an associated Sturm–Liouville problem. The eigenvalues 1, <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/2⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠, ... form the (musical) <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a>.</figcaption></figure> In the theory of <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a>, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the <a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville problem</a> arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in <a href="/wiki/Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a>.<a href="#cite_note-39">[38]</a> The problem is a differential equation of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=\lambda w(x)y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo>=</mo> <mi>λ</mi> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=\lambda w(x)y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4e93c4082311667644e8554f78774b963ffb0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.492ex; height:6.176ex;" alt="{\displaystyle -{\frac {\mathrm {d} }{\mathrm {d} x}}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=\lambda w(x)y}"> for an unknown function y on an interval [a, b], satisfying general homogeneous <a href="/wiki/Robin_boundary_conditions" class="mw-redirect" title="Robin boundary conditions">Robin boundary conditions</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\alpha y(a)+\alpha 'y'(a)&=0\\\beta y(b)+\beta 'y'(b)&=0\,.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>α</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>α</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>β</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\alpha y(a)+\alpha 'y'(a)&=0\\\beta y(b)+\beta 'y'(b)&=0\,.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992667418f6db4ca72e98721e1d698be8a90a7e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.046ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\alpha y(a)+\alpha 'y'(a)&=0\\\beta y(b)+\beta 'y'(b)&=0\,.\end{cases}}}"> The functions p, q, and w are given in advance, and the problem is to find the function y and constants λ for which the equation has a solution. The problem only has solutions for certain values of λ, called eigenvalues of the system, and this is a consequence of the spectral theorem for <a href="/wiki/Compact_operator" title="Compact operator">compact operators</a> applied to the <a href="/wiki/Integral_operator" title="Integral operator">integral operator</a> defined by the <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a> for the system. Furthermore, another consequence of this general result is that the eigenvalues λ of the system can be arranged in an increasing sequence tending to infinity.<a href="#cite_note-40">[39]</a><a href="#cite_note-41">[nb 2]</a> <div class="mw-heading mw-heading3"><h3 id="Partial_differential_equations">Partial differential equations</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=14" title="Edit section: Partial differential equations">edit</a>]</div> Hilbert spaces form a basic tool in the study of <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>.<a href="#cite_note-BeJoSc81-32">[31]</a> For many classes of partial differential equations, such as linear <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic equations</a>, it is possible to consider a generalized solution (known as a <a href="/wiki/Weak_derivative" title="Weak derivative">weak</a> solution) by enlarging the class of functions. Many weak formulations involve the class of <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev functions</a>, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem, the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the <a href="/wiki/Lax%E2%80%93Milgram_theorem" class="mw-redirect" title="Lax–Milgram theorem">Lax–Milgram theorem</a>. This strategy forms the rudiment of the <a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a> (a <a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a>) for numerical solution of partial differential equations.<a href="#cite_note-42">[40]</a> A typical example is the <a href="/wiki/Poisson_equation" class="mw-redirect" title="Poisson equation">Poisson equation</a> −Δu = g with <a href="/wiki/Dirichlet_boundary_conditions" class="mw-redirect" title="Dirichlet boundary conditions">Dirichlet boundary conditions</a> in a bounded domain Ω in R2. The weak formulation consists of finding a function u such that, for all continuously differentiable functions v in Ω vanishing on the boundary: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\Omega }\nabla u\cdot \nabla v=\int _{\Omega }gv\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>g</mi> <mi>v</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\nabla u\cdot \nabla v=\int _{\Omega }gv\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8bcf957f73f0f6f2b6c40537741271a5923c38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.581ex; height:5.676ex;" alt="{\displaystyle \int _{\Omega }\nabla u\cdot \nabla v=\int _{\Omega }gv\,.}"> This can be recast in terms of the Hilbert space H1 0(Ω) consisting of functions u such that u, along with its weak partial derivatives, are square integrable on Ω, and vanish on the boundary. The question then reduces to finding u in this space such that for all v in this space <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(u,v)=b(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(u,v)=b(v)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/386fc8bc772c1d357db4e4549e05ad79174118df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.563ex; height:2.843ex;" alt="{\displaystyle a(u,v)=b(v)}"> where a is a continuous <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a>, and b is a continuous <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a>, given respectively by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(u,v)=\int _{\Omega }\nabla u\cdot \nabla v,\quad b(v)=\int _{\Omega }gv\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>g</mi> <mi>v</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(u,v)=\int _{\Omega }\nabla u\cdot \nabla v,\quad b(v)=\int _{\Omega }gv\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda62b2fa76285d30b8ca17da5714caa584b0da4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.501ex; height:5.676ex;" alt="{\displaystyle a(u,v)=\int _{\Omega }\nabla u\cdot \nabla v,\quad b(v)=\int _{\Omega }gv\,.}"> Since the Poisson equation is <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic</a>, it follows from Poincaré's inequality that the bilinear form a is <a href="/wiki/Coercive_function" title="Coercive function">coercive</a>. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.<a href="#cite_note-43">[41]</a> Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to <a href="/wiki/Parabolic_partial_differential_equation" title="Parabolic partial differential equation">parabolic partial differential equations</a> and certain <a href="/wiki/Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">hyperbolic partial differential equations</a>.<a href="#cite_note-44">[42]</a> <div class="mw-heading mw-heading3"><h3 id="Ergodic_theory">Ergodic theory</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=15" title="Edit section: Ergodic theory">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:BunimovichStadium.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/BunimovichStadium.svg/220px-BunimovichStadium.svg.png" decoding="async" width="220" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/BunimovichStadium.svg/330px-BunimovichStadium.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/BunimovichStadium.svg/440px-BunimovichStadium.svg.png 2x" data-file-width="663" data-file-height="324" /></a><figcaption>The path of a <a href="/wiki/Dynamical_billiards" title="Dynamical billiards">billiard</a> ball in the <a href="/wiki/Bunimovich_stadium" class="mw-redirect" title="Bunimovich stadium">Bunimovich stadium</a> is described by an ergodic <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>.</figcaption></figure> The field of <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a> is the study of the long-term behavior of <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic</a> <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>. The protypical case of a field that ergodic theory applies to is <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable. The <a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">laws of thermodynamics</a> are assertions about such average behavior. In particular, one formulation of the <a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">zeroth law of thermodynamics</a> asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of <a href="/wiki/Temperature" title="Temperature">temperature</a>.<a href="#cite_note-45">[43]</a> An ergodic dynamical system is one for which, apart from the energy—measured by the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>—there are no other functionally independent <a href="/wiki/Conserved_quantities" class="mw-redirect" title="Conserved quantities">conserved quantities</a> on the <a href="/wiki/Phase_space" title="Phase space">phase space</a>. More explicitly, suppose that the energy E is fixed, and let ΩE be the subset of the phase space consisting of all states of energy E (an energy surface), and let Tt denote the evolution operator on the phase space. The dynamical system is ergodic if every invariant measurable functions on ΩE is constant <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>.<a href="#cite_note-46">[44]</a> An invariant function f is one for which <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(T_{t}w)=f(w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T_{t}w)=f(w)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf2a493c10cb88b35c1d4a145926a7ff2cd8842" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.786ex; height:2.843ex;" alt="{\displaystyle f(T_{t}w)=f(w)}"> for all w on ΩE and all time t. <a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem</a> implies that there exists a <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure</a> μ on the energy surface that is invariant under the <a href="/wiki/Time_translation" class="mw-redirect" title="Time translation">time translation</a>. As a result, time translation is a <a href="/wiki/Unitary_transformation" title="Unitary transformation">unitary transformation</a> of the Hilbert space L2(ΩE, μ) consisting of square-integrable functions on the energy surface ΩE with respect to the inner product <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle f,g\right\rangle _{L^{2}\left(\Omega _{E},\mu \right)}=\int _{E}f{\bar {g}}\,\mathrm {d} \mu \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>⟨</mo> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>,</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle f,g\right\rangle _{L^{2}\left(\Omega _{E},\mu \right)}=\int _{E}f{\bar {g}}\,\mathrm {d} \mu \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/decbac50d55419a511d0372b6dfad9baa58a20f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.41ex; height:5.676ex;" alt="{\displaystyle \left\langle f,g\right\rangle _{L^{2}\left(\Omega _{E},\mu \right)}=\int _{E}f{\bar {g}}\,\mathrm {d} \mu \,.}"> The von Neumann mean ergodic theorem<a href="#cite_note-von_Neumann_1932-25">[24]</a> states the following: <ul><li>If Ut is a (strongly continuous) one-parameter <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> of unitary operators on a Hilbert space H, and P is the orthogonal projection onto the space of common fixed points of Ut, {x ∈H | Utx = x, ∀t > 0}, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Px=\lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}U_{t}x\,\mathrm {d} t\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>x</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Px=\lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}U_{t}x\,\mathrm {d} t\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58abae503a6878f01de9a3c3517ae8c94ecfbd73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.4ex; height:6.176ex;" alt="{\displaystyle Px=\lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}U_{t}x\,\mathrm {d} t\,.}"></li></ul> For an ergodic system, the fixed set of the time evolution consists only of the constant functions, so the ergodic theorem implies the following:<a href="#cite_note-47">[45]</a> for any function f ∈ L2(ΩE, μ), <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underset {T\to \infty }{L^{2}-\lim }}{\frac {1}{T}}\int _{0}^{T}f(T_{t}w)\,\mathrm {d} t=\int _{\Omega _{E}}f(y)\,\mathrm {d} \mu (y)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo movablelimits="true" form="prefix">lim</mo> </mrow> <mrow> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underset {T\to \infty }{L^{2}-\lim }}{\frac {1}{T}}\int _{0}^{T}f(T_{t}w)\,\mathrm {d} t=\int _{\Omega _{E}}f(y)\,\mathrm {d} \mu (y)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b23f47d65d6729b56ee0921737f1840150206835" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.966ex; height:6.509ex;" alt="{\displaystyle {\underset {T\to \infty }{L^{2}-\lim }}{\frac {1}{T}}\int _{0}^{T}f(T_{t}w)\,\mathrm {d} t=\int _{\Omega _{E}}f(y)\,\mathrm {d} \mu (y)\,.}"> That is, the long time average of an observable f is equal to its expectation value over an energy surface. <div class="mw-heading mw-heading3"><h3 id="Fourier_analysis">Fourier analysis</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=16" title="Edit section: Fourier analysis">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sawtooth_Fourier_Analysys.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Sawtooth_Fourier_Analysys.svg/220px-Sawtooth_Fourier_Analysys.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Sawtooth_Fourier_Analysys.svg/330px-Sawtooth_Fourier_Analysys.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Sawtooth_Fourier_Analysys.svg/440px-Sawtooth_Fourier_Analysys.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top)</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Harmoniki.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Harmoniki.png/220px-Harmoniki.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Harmoniki.png/330px-Harmoniki.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Harmoniki.png/440px-Harmoniki.png 2x" data-file-width="1280" data-file-height="721" /></a><figcaption><a href="/wiki/Spherical_harmonics" title="Spherical harmonics">Spherical harmonics</a>, an orthonormal basis for the Hilbert space of square-integrable functions on the sphere, shown graphed along the radial direction</figcaption></figure> One of the basic goals of <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> is to decompose a function into a (possibly infinite) <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of given basis functions: the associated <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. The classical Fourier series associated to a function f defined on the interval [0, 1] is a series of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{2\pi in\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>n</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{2\pi in\theta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2147afcbe4d3a488c888213ccba7a7aff0b3f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.426ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{2\pi in\theta }}"> where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=\int _{0}^{1}f(\theta )\;\!e^{-2\pi in\theta }\,\mathrm {d} \theta \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>n</mi> <mi>θ</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=\int _{0}^{1}f(\theta )\;\!e^{-2\pi in\theta }\,\mathrm {d} \theta \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929652493606593c89b5d1e2831e236bd4d59912" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.375ex; height:6.176ex;" alt="{\displaystyle a_{n}=\int _{0}^{1}f(\theta )\;\!e^{-2\pi in\theta }\,\mathrm {d} \theta \,.}"> The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The basis functions are sine waves with wavelengths <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠λ/n⁠ (for integer n) shorter than the wavelength λ of the sawtooth itself (except for n = 1, the fundamental wave). A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f. Hilbert space methods provide one possible answer to this question.<a href="#cite_note-48">[46]</a> The functions en(θ) = e2πinθ form an orthogonal basis of the Hilbert space L2([0, 1]). Consequently, any square-integrable function can be expressed as a series <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta )=\sum _{n}a_{n}e_{n}(\theta )\,,\quad a_{n}=\langle f,e_{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta )=\sum _{n}a_{n}e_{n}(\theta )\,,\quad a_{n}=\langle f,e_{n}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180ca7e1463bb5f755ba1377633d8643b9b9d312" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.383ex; height:5.509ex;" alt="{\displaystyle f(\theta )=\sum _{n}a_{n}e_{n}(\theta )\,,\quad a_{n}=\langle f,e_{n}\rangle }"> and, moreover, this series converges in the Hilbert space sense (that is, in the <a href="/wiki/Mean_convergence" class="mw-redirect" title="Mean convergence">L2 mean</a>). The problem can also be studied from the abstract point of view: every Hilbert space has an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space.<a href="#cite_note-49">[47]</a> The abstraction is especially useful when it is more natural to use different basis functions for a space such as L2([0, 1]). In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into <a href="/wiki/Orthogonal_polynomials" title="Orthogonal polynomials">orthogonal polynomials</a> or <a href="/wiki/Wavelet" title="Wavelet">wavelets</a> for instance,<a href="#cite_note-50">[48]</a> and in higher dimensions into <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>.<a href="#cite_note-51">[49]</a> For instance, if en are any orthonormal basis functions of L2[0, 1], then a given function in L2[0, 1] can be approximated as a finite linear combination<a href="#cite_note-52">[50]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\approx f_{n}(x)=a_{1}e_{1}(x)+a_{2}e_{2}(x)+\cdots +a_{n}e_{n}(x)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\approx f_{n}(x)=a_{1}e_{1}(x)+a_{2}e_{2}(x)+\cdots +a_{n}e_{n}(x)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7942e7d3c2d9843987b8382eeae55b7f1f1f04b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.4ex; height:2.843ex;" alt="{\displaystyle f(x)\approx f_{n}(x)=a_{1}e_{1}(x)+a_{2}e_{2}(x)+\cdots +a_{n}e_{n}(x)\,.}"> The coefficients {aj} are selected to make the magnitude of the difference ‖f − fn‖2 as small as possible. Geometrically, the <a href="#Best_approximation">best approximation</a> is the <a href="#Orthogonal_complements_and_projections">orthogonal projection</a> of f onto the subspace consisting of all linear combinations of the {ej}, and can be calculated by<a href="#cite_note-53">[51]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}=\int _{0}^{1}{\overline {e_{j}(x)}}f(x)\,\mathrm {d} x\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j}=\int _{0}^{1}{\overline {e_{j}(x)}}f(x)\,\mathrm {d} x\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6880852f6c9f4506bca5fc28e0dfdf088fa9ae19" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.851ex; height:6.176ex;" alt="{\displaystyle a_{j}=\int _{0}^{1}{\overline {e_{j}(x)}}f(x)\,\mathrm {d} x\,.}"> That this formula minimizes the difference ‖f − fn‖2 is a consequence of <a href="#Bessel's_inequality_and_Parseval's_formula">Bessel's inequality and Parseval's formula</a>. In various applications to physical problems, a function can be decomposed into physically meaningful <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of a <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> (typically the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a>): this forms the foundation for the spectral study of functions, in reference to the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectrum</a> of the differential operator.<a href="#cite_note-54">[52]</a> A concrete physical application involves the problem of <a href="/wiki/Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">hearing the shape of a drum</a>: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself?<a href="#cite_note-55">[53]</a> The mathematical formulation of this question involves the <a href="/wiki/Dirichlet_eigenvalue" title="Dirichlet eigenvalue">Dirichlet eigenvalues</a> of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string. <a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> also underlies certain aspects of the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of a function. Whereas Fourier analysis decomposes a function defined on a <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact set</a> into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the <a href="/wiki/Continuous_spectrum" class="mw-redirect" title="Continuous spectrum">continuous spectrum</a> of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the <a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a>, that asserts that it is an <a href="/wiki/Isometry" title="Isometry">isometry</a> of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> (since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by the <a href="/wiki/Plancherel_theorem_for_spherical_functions" title="Plancherel theorem for spherical functions">Plancherel theorem for spherical functions</a> occurring in <a href="/wiki/Noncommutative_harmonic_analysis" title="Noncommutative harmonic analysis">noncommutative harmonic analysis</a>. <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=17" title="Edit section: Quantum mechanics">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:HAtomOrbitals.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/HAtomOrbitals.png/220px-HAtomOrbitals.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/HAtomOrbitals.png 1.5x" data-file-width="316" data-file-height="316" /></a><figcaption>The <a href="/wiki/Atomic_orbital" title="Atomic orbital">orbitals</a> of an <a href="/wiki/Electron" title="Electron">electron</a> in a <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a> are <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> of the <a href="/wiki/Energy_(physics)" class="mw-redirect" title="Energy (physics)">energy</a>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement in quantum mechanics</a></div> In the mathematically rigorous formulation of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, developed by <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>,<a href="#cite_note-56">[54]</a> the possible states (more precisely, the <a href="/wiki/Pure_state" class="mw-redirect" title="Pure state">pure states</a>) of a quantum mechanical system are represented by <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a> (called state vectors) residing in a complex separable Hilbert space, known as the <a href="/wiki/Quantum_state_space" title="Quantum state space">state space</a>, well defined up to a complex number of norm 1 (the <a href="/wiki/Phase_factor" title="Phase factor">phase factor</a>). In other words, the possible states are points in the <a href="/wiki/Projective_space" title="Projective space">projectivization</a> of a Hilbert space, usually called the <a href="/wiki/Complex_projective_space" title="Complex projective space">complex projective space</a>. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all <a href="/wiki/Square-integrable" class="mw-redirect" title="Square-integrable">square-integrable</a> functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of <a href="/wiki/Spinors_in_three_dimensions" title="Spinors in three dimensions">spinors</a>. Each observable is represented by a <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint</a> <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> acting on the state space. Each eigenstate of an observable corresponds to an <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a> of the operator, and the associated <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> corresponds to the value of the observable in that eigenstate.<a href="#cite_note-57">[55]</a> The inner product between two state vectors is a complex number known as a <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitude</a>. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the probability amplitudes between the initial and final states.<a href="#cite_note-58">[56]</a> The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.<a href="#cite_note-59">[57]</a> For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by <a href="/wiki/Density_matrix" title="Density matrix">density matrices</a>: self-adjoint operators of <a href="/wiki/Trace_of_a_matrix" class="mw-redirect" title="Trace of a matrix">trace</a> one on a Hilbert space.<a href="#cite_note-60">[58]</a> Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a <a href="/wiki/Positive_operator_valued_measure" class="mw-redirect" title="Positive operator valued measure">positive operator valued measure</a>. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.<a href="#cite_note-61">[59]</a> <div class="mw-heading mw-heading3"><h3 id="Probability_theory">Probability theory</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=18" title="Edit section: Probability theory">edit</a>]</div> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, Hilbert spaces also have diverse applications. Here a fundamental Hilbert space is the space of <a href="/wiki/Random_variable" title="Random variable">random variables</a> on a given <a href="/wiki/Probability_space" title="Probability space">probability space</a>, having class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> (finite first and second <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a>). A common operation in statistics is that of centering a random variable by subtracting its <a href="/wiki/Expected_value" title="Expected value">expectation</a>. Thus if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a random variable, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-E(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X-E(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0abbac80451543e002ee8914298f66c8cd53a8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:2.843ex;" alt="{\displaystyle X-E(X)}"> is its centering. In the Hilbert space view, this is the orthogonal projection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> onto the <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of the expectation operator, which a <a href="/wiki/Continuous_linear_functional" class="mw-redirect" title="Continuous linear functional">continuous linear functional</a> on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace. The <a href="/wiki/Conditional_expectation" title="Conditional expectation">conditional expectation</a> has a natural interpretation in the Hilbert space.<a href="#cite_note-62">[60]</a> Suppose that a probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,P,{\mathcal {B}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>P</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,P,{\mathcal {B}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2a7e94a34894aec6994e0834db493394c87aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.844ex; height:2.843ex;" alt="{\displaystyle (\Omega ,P,{\mathcal {B}})}"> is given, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}"> is a <a href="/wiki/Sigma_algebra" class="mw-redirect" title="Sigma algebra">sigma algebra</a> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> on the measure space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {B}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {B}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c71c3b681b11d672fb4712a121dc140ff810ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.065ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {B}})}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\leq {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\leq {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28298e2f52274e8c0bf0e03dbe4e40f15691e2a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.568ex; height:2.343ex;" alt="{\displaystyle {\mathcal {F}}\leq {\mathcal {B}}}"> is a sigma subalgebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}">, then the conditional expectation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X|{\mathcal {F}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[X|{\mathcal {F}}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda70bf9ef6fa0e4a2acbc19572fc775644a4225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.623ex; height:2.843ex;" alt="{\displaystyle E[X|{\mathcal {F}}]}"> is the orthogonal projection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> onto the subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Omega ,P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\Omega ,P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fcb896d24e39cee28d95a2483f448d3fa231ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.904ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Omega ,P)}"> consisting of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">-measurable functions. If the random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Omega ,P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\Omega ,P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fcb896d24e39cee28d95a2483f448d3fa231ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.904ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Omega ,P)}"> is independent of the sigma algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> then conditional expectation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(X|{\mathcal {F}})=E(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X|{\mathcal {F}})=E(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c35b1a7858cae37cc026b8844fa536f955cf648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.802ex; height:2.843ex;" alt="{\displaystyle E(X|{\mathcal {F}})=E(X)}">, i.e., its projection onto the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">-measurable functions is constant. Equivalently, the projection of its centering is zero. In particular, if two random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> (in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Omega ,P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\Omega ,P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fcb896d24e39cee28d95a2483f448d3fa231ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.904ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Omega ,P)}">) are independent, then the centered random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-E(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X-E(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0abbac80451543e002ee8914298f66c8cd53a8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:2.843ex;" alt="{\displaystyle X-E(X)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y-E(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y-E(Y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d3edaee1c70a145e670d5de5d609b729167ab65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.972ex; height:2.843ex;" alt="{\displaystyle Y-E(Y)}"> are orthogonal. (This means that the two variables have zero <a href="/wiki/Covariance" title="Covariance">covariance</a>: they are <a href="/wiki/Uncorrelated" class="mw-redirect" title="Uncorrelated">uncorrelated</a>.) In that case, the Pythagorean theorem in the kernel of the expectation operator implies that the <a href="/wiki/Variance" title="Variance">variances</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"> satisfy the identity: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (X+Y)=\operatorname {Var} (X)+\operatorname {Var} (Y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X+Y)=\operatorname {Var} (X)+\operatorname {Var} (Y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edda73fedaf279c028c7bbf77cd3cb044e301e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.812ex; height:2.843ex;" alt="{\displaystyle \operatorname {Var} (X+Y)=\operatorname {Var} (X)+\operatorname {Var} (Y),}"> sometimes called the Pythagorean theorem of statistics, and is of importance in <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a>.<a href="#cite_note-63">[61]</a> As <a href="#CITEREFStapleton1995">Stapleton (1995)</a> puts it, "the <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a> may be viewed as the decomposition of the squared length of a vector into the sum of the squared lengths of several vectors, using the Pythagorean Theorem." The theory of <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingales</a> can be formulated in Hilbert spaces. A martingale in a Hilbert space is a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71a0bff46c1d8ad13a271387dcf9c11fb30cee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.559ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\dots }"> of elements of a Hilbert space such that, for each n, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"> is the orthogonal projection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e76103271a353337f6b3c697209cb19359807df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.649ex; height:2.009ex;" alt="{\displaystyle x_{n+1}}"> onto the linear hull of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{n}}">.<a href="#cite_note-64">[62]</a> If the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"> are random variables, this reproduces the usual definition of a (discrete) martingale: the expectation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e76103271a353337f6b3c697209cb19359807df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.649ex; height:2.009ex;" alt="{\displaystyle x_{n+1}}">, conditioned on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{n}}">, is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}">. Hilbert spaces are also used throughout the foundations of the <a href="/wiki/It%C3%B4_calculus" title="Itô calculus">Itô calculus</a>.<a href="#cite_note-65">[63]</a> To any square-integrable <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a>, it is possible to associate a Hilbert norm on the space of equivalence classes of <a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">progressively measurable processes</a> with respect to the martingale (using the <a href="/wiki/Quadratic_variation" title="Quadratic variation">quadratic variation</a> of the martingale as the measure). The <a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a> can be constructed by first defining it for <a href="/w/index.php?title=Simple_process&action=edit&redlink=1" class="new" title="Simple process (page does not exist)">simple processes</a>, and then exploiting their density in the Hilbert space. A noteworthy result is then the <a href="/wiki/It%C3%B4_isometry" title="Itô isometry">Itô isometry</a>, which attests that for any martingale M having quadratic variation measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\langle M\rangle _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>M</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\langle M\rangle _{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8ebcf606010792fcbc0ce38340f2f2f774daa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.293ex; height:2.843ex;" alt="{\displaystyle d\langle M\rangle _{t}}">, and any progressively measurable process H: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[\left(\int _{0}^{t}H_{s}dM_{s}\right)^{2}\right]=E\left[\int _{0}^{t}H_{s}^{2}d\langle M\rangle _{s}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>d</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>M</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[\left(\int _{0}^{t}H_{s}dM_{s}\right)^{2}\right]=E\left[\int _{0}^{t}H_{s}^{2}d\langle M\rangle _{s}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ddc67bd90763bdda507f2bfd375f0106f921894" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.455ex; height:7.509ex;" alt="{\displaystyle E\left[\left(\int _{0}^{t}H_{s}dM_{s}\right)^{2}\right]=E\left[\int _{0}^{t}H_{s}^{2}d\langle M\rangle _{s}\right]}"> whenever the expectation on the right-hand side is finite. A deeper application of Hilbert spaces that is especially important in the theory of <a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian processes</a> is an attempt, due to <a href="/wiki/Leonard_Gross" title="Leonard Gross">Leonard Gross</a> and others, to make sense of certain formal integrals over infinite dimensional spaces like the <a href="/wiki/Feynman_path_integral" class="mw-redirect" title="Feynman path integral">Feynman path integral</a> from <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. The problem with integral like this is that there is no <a href="/wiki/Infinite_dimensional_Lebesgue_measure" class="mw-redirect" title="Infinite dimensional Lebesgue measure">infinite dimensional Lebesgue measure</a>. The notion of an <a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">abstract Wiener space</a> allows one to construct a measure on a Banach space B that contains a Hilbert space H, called the <a href="/wiki/Cameron%E2%80%93Martin_space" class="mw-redirect" title="Cameron–Martin space">Cameron–Martin space</a>, as a dense subset, out of a finitely additive cylinder set measure on H. The resulting measure on B is countably additive and invariant under translation by elements of H, and this provides a mathematically rigorous way of thinking of the <a href="/wiki/Wiener_measure" class="mw-redirect" title="Wiener measure">Wiener measure</a> as a Gaussian measure on the Sobolev space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{1}([0,\infty ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{1}([0,\infty ))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac69384c0c142a8bcb92dfa23538dd4f3c8f1f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.039ex; height:3.176ex;" alt="{\displaystyle H^{1}([0,\infty ))}">.<a href="#cite_note-66">[64]</a> <div class="mw-heading mw-heading3"><h3 id="Color_perception">Color perception</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=19" title="Edit section: Color perception">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Color_vision#Mathematics_of_color_perception" title="Color vision">Color vision § Mathematics of color perception</a></div> Any true physical color can be represented by a combination of pure <a href="/wiki/Spectral_color" title="Spectral color">spectral colors</a>. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans have <a href="/wiki/Trichromacy" title="Trichromacy">three types of cone cells</a> for color perception, so the perceivable colors can be represented by 3-dimensional Euclidean space. The many-to-one linear mapping from the Hilbert space of physical colors to the Euclidean space of human perceivable colors explains why many distinct physical colors may be perceived by humans to be identical (e.g., pure yellow light versus a mix of red and green light, see <a href="/wiki/Metamerism_(color)" title="Metamerism (color)">metamerism</a>).<a href="#cite_note-67">[65]</a><a href="#cite_note-68">[66]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=20" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Pythagorean_identity">Pythagorean identity</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=21" title="Edit section: Pythagorean identity">edit</a>]</div> Two vectors u and v in a Hilbert space H are orthogonal when ⟨u, v⟩ = 0. The notation for this is u ⊥ v. More generally, when S is a subset in H, the notation u ⊥ S means that u is orthogonal to every element from S. When u and v are orthogonal, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|^{2}=\langle u+v,u+v\rangle =\langle u,u\rangle +2\,\operatorname {Re} \langle u,v\rangle +\langle v,v\rangle =\|u\|^{2}+\|v\|^{2}\,.}"> <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> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|u+v\|^{2}=\langle u+v,u+v\rangle =\langle u,u\rangle +2\,\operatorname {Re} \langle u,v\rangle +\langle v,v\rangle =\|u\|^{2}+\|v\|^{2}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18db6083346d3b1f8e925f0d2d6a5a1cf02d288" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.763ex; height:3.176ex;" alt="{\displaystyle \|u+v\|^{2}=\langle u+v,u+v\rangle =\langle u,u\rangle +2\,\operatorname {Re} \langle u,v\rangle +\langle v,v\rangle =\|u\|^{2}+\|v\|^{2}\,.}"> By induction on n, this is extended to any family u1, ..., un of n orthogonal vectors, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|u_{1}+\cdots +u_{n}\right\|^{2}=\left\|u_{1}\right\|^{2}+\cdots +\left\|u_{n}\right\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mi>u</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> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|u_{1}+\cdots +u_{n}\right\|^{2}=\left\|u_{1}\right\|^{2}+\cdots +\left\|u_{n}\right\|^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8a2ecc3d4b7abb352ce0887b1eec38f97295f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.555ex; height:3.343ex;" alt="{\displaystyle \left\|u_{1}+\cdots +u_{n}\right\|^{2}=\left\|u_{1}\right\|^{2}+\cdots +\left\|u_{n}\right\|^{2}.}"> Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series.<a href="#cite_note-69">[67]</a> A series Σuk of orthogonal vectors converges in H if and only if the series of squares of norms converges, and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Biggl \|}\sum _{k=0}^{\infty }u_{k}{\Biggr \|}^{2}=\sum _{k=0}^{\infty }\left\|u_{k}\right\|^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo symmetric="true" maxsize="2.470em" minsize="2.470em">‖</mo> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo symmetric="true" maxsize="2.470em" minsize="2.470em">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Biggl \|}\sum _{k=0}^{\infty }u_{k}{\Biggr \|}^{2}=\sum _{k=0}^{\infty }\left\|u_{k}\right\|^{2}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2cbc044bcb2f7757753e87855fe3ff07aae355" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.599ex; height:7.343ex;" alt="{\displaystyle {\Biggl \|}\sum _{k=0}^{\infty }u_{k}{\Biggr \|}^{2}=\sum _{k=0}^{\infty }\left\|u_{k}\right\|^{2}\,.}"> Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken. <div class="mw-heading mw-heading3"><h3 id="Parallelogram_identity_and_polarization">Parallelogram identity and polarization</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=22" title="Edit section: Parallelogram identity and polarization">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Color_parallelogram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Color_parallelogram.svg/220px-Color_parallelogram.svg.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Color_parallelogram.svg/330px-Color_parallelogram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Color_parallelogram.svg/440px-Color_parallelogram.svg.png 2x" data-file-width="255" data-file-height="175" /></a><figcaption>Geometrically, the parallelogram identity asserts that AC2 + BD2 = 2(AB2 + AD2). In words, the sum of the squares of the diagonals is twice the sum of the squares of any two adjacent sides.</figcaption></figure> By definition, every Hilbert space is also a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. Furthermore, in every Hilbert space the following <a href="/wiki/Parallelogram_identity" class="mw-redirect" title="Parallelogram identity">parallelogram identity</a> holds:<a href="#cite_note-70">[68]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|^{2}+\|u-v\|^{2}=2{\bigl (}\|u\|^{2}+\|v\|^{2}{\bigr )}\,.}"> <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> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo>−</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|u+v\|^{2}+\|u-v\|^{2}=2{\bigl (}\|u\|^{2}+\|v\|^{2}{\bigr )}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99f0e0121d10601ed32a1acf93b491870eef3bd3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.675ex; height:3.343ex;" alt="{\displaystyle \|u+v\|^{2}+\|u-v\|^{2}=2{\bigl (}\|u\|^{2}+\|v\|^{2}{\bigr )}\,.}"> Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the <a href="/wiki/Polarization_identity" title="Polarization identity">polarization identity</a>.<a href="#cite_note-71">[69]</a> For real Hilbert spaces, the polarization identity is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle ={\tfrac {1}{4}}{\bigl (}\|u+v\|^{2}-\|u-v\|^{2}{\bigr )}\,.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo>−</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle ={\tfrac {1}{4}}{\bigl (}\|u+v\|^{2}-\|u-v\|^{2}{\bigr )}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbddb1a3d65ef0456f237dcad6a74b6ce8954e02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.415ex; height:3.509ex;" alt="{\displaystyle \langle u,v\rangle ={\tfrac {1}{4}}{\bigl (}\|u+v\|^{2}-\|u-v\|^{2}{\bigr )}\,.}"> For complex Hilbert spaces, it is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle ={\tfrac {1}{4}}{\bigl (}\|u+v\|^{2}-\|u-v\|^{2}+i\|u+iv\|^{2}-i\|u-iv\|^{2}{\bigr )}\,.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo>−</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <mo>−</mo> <mi>i</mi> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle ={\tfrac {1}{4}}{\bigl (}\|u+v\|^{2}-\|u-v\|^{2}+i\|u+iv\|^{2}-i\|u-iv\|^{2}{\bigr )}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e178f0bd9f69910d6d29b3159da2a40fc17481fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.659ex; height:3.509ex;" alt="{\displaystyle \langle u,v\rangle ={\tfrac {1}{4}}{\bigl (}\|u+v\|^{2}-\|u-v\|^{2}+i\|u+iv\|^{2}-i\|u-iv\|^{2}{\bigr )}\,.}"> The parallelogram law implies that any Hilbert space is a <a href="/wiki/Uniformly_convex_Banach_space" class="mw-redirect" title="Uniformly convex Banach space">uniformly convex Banach space</a>.<a href="#cite_note-72">[70]</a> <div class="mw-heading mw-heading3"><h3 id="Best_approximation">Best approximation</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=23" title="Edit section: Best approximation">edit</a>]</div> This subsection employs the <a href="/wiki/Hilbert_projection_theorem" title="Hilbert projection theorem">Hilbert projection theorem</a>. If C is a non-empty closed convex subset of a Hilbert space H and x a point in H, there exists a unique point y ∈ C that minimizes the distance between x and points in C,<a href="#cite_note-73">[71]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in C\,,\quad \|x-y\|=\operatorname {dist} (x,C)=\min {\bigl \{}\|x-z\|\mathrel {\big |} z\in C{\bigr \}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>C</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mi>dist</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mi>z</mi> <mo>∈</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in C\,,\quad \|x-y\|=\operatorname {dist} (x,C)=\min {\bigl \{}\|x-z\|\mathrel {\big |} z\in C{\bigr \}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4e7ee5ff6bed2e474dc7061e2a68c1ee67522f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.275ex; height:3.176ex;" alt="{\displaystyle y\in C\,,\quad \|x-y\|=\operatorname {dist} (x,C)=\min {\bigl \{}\|x-z\|\mathrel {\big |} z\in C{\bigr \}}\,.}"> This is equivalent to saying that there is a point with minimal norm in the translated convex set D = C − x. The proof consists in showing that every minimizing sequence (dn) ⊂ D is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point in D that has minimal norm. More generally, this holds in any uniformly convex Banach space.<a href="#cite_note-74">[72]</a> When this result is applied to a closed subspace F of H, it can be shown that the point y ∈ F closest to x is characterized by<a href="#cite_note-75">[73]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in F\,,\quad x-y\perp F\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>F</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>⊥</mo> <mi>F</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in F\,,\quad x-y\perp F\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51abff2fb0fa9975d7b21d7945b9ec722369b5d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.679ex; height:2.509ex;" alt="{\displaystyle y\in F\,,\quad x-y\perp F\,.}"> This point y is the orthogonal projection of x onto F, and the mapping PF : x → y is linear (see <a href="#Orthogonal_complements_and_projections">Orthogonal complements and projections</a>). This result is especially significant in <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>, especially <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, where it forms the basis of <a href="/wiki/Least_squares" title="Least squares">least squares</a> methods.<a href="#cite_note-76">[74]</a> In particular, when F is not equal to H, one can find a nonzero vector v orthogonal to F (select x ∉ F and v = x − y). A very useful criterion is obtained by applying this observation to the closed subspace F generated by a subset S of H. <dl><dd>A subset S of H spans a dense vector subspace if (and only if) the vector 0 is the sole vector v ∈ H orthogonal to S.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Duality">Duality</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=24" title="Edit section: Duality">edit</a>]</div> The <a href="/wiki/Continuous_dual_space" class="mw-redirect" title="Continuous dual space">dual space</a> H* is the space of all <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> linear functions from the space H into the base field. It carries a natural norm, defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\varphi \|=\sup _{\|x\|=1,x\in H}|\varphi (x)|\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>φ</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <mi>H</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\varphi \|=\sup _{\|x\|=1,x\in H}|\varphi (x)|\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ca1b74cb8e720c7d8ebc2a59d67b6427d479ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.955ex; height:5.009ex;" alt="{\displaystyle \|\varphi \|=\sup _{\|x\|=1,x\in H}|\varphi (x)|\,.}"> This norm satisfies the <a href="/wiki/Parallelogram_law" title="Parallelogram law">parallelogram law</a>, and so the dual space is also an inner product space where this inner product can be defined in terms of this dual norm by using the <a href="/wiki/Polarization_identity" title="Polarization identity">polarization identity</a>. The dual space is also complete so it is a Hilbert space in its own right. If e• = (ei)i ∈ I is a complete orthonormal basis for H then the inner product on the dual space of any two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in H^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in H^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35514ab9493e958f4c5de4ebfbaf2c7561222492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.427ex; height:2.676ex;" alt="{\displaystyle f,g\in H^{*}}"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle _{H^{*}}=\sum _{i\in I}f(e_{i}){\overline {g(e_{i})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle _{H^{*}}=\sum _{i\in I}f(e_{i}){\overline {g(e_{i})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad405e9182a274919f6018875a65f20132ced37f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.524ex; height:6.009ex;" alt="{\displaystyle \langle f,g\rangle _{H^{*}}=\sum _{i\in I}f(e_{i}){\overline {g(e_{i})}}}"> where all but countably many of the terms in this series are zero. The <a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a> affords a convenient description of the dual space. To every element u of H, there is a unique element φu of H*, defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{u}(x)=\langle x,u\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{u}(x)=\langle x,u\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb78bcc918808e9e933a0096a2af8211f15a029" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.433ex; height:2.843ex;" alt="{\displaystyle \varphi _{u}(x)=\langle x,u\rangle }"> where moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|\varphi _{u}\right\|=\left\|u\right\|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mi>u</mi> <mo symmetric="true">‖</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\varphi _{u}\right\|=\left\|u\right\|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb85d4871783b5ce911c1466616a910ea1422cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.805ex; height:2.843ex;" alt="{\displaystyle \left\|\varphi _{u}\right\|=\left\|u\right\|.}"> The Riesz representation theorem states that the map from H to H* defined by u ↦ φu is <a href="/wiki/Surjective_map" class="mw-redirect" title="Surjective map">surjective</a>, which makes this map an <a href="/wiki/Isometry" title="Isometry">isometric</a> <a href="/wiki/Antilinear_map" title="Antilinear map">antilinear</a> isomorphism.<a href="#cite_note-77">[75]</a> So to every element φ of the dual H* there exists one and only one uφ in H such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,u_{\varphi }\rangle =\varphi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,u_{\varphi }\rangle =\varphi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e56abca14ae484d69e5eaeaee5d875b3e1f24b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.567ex; height:3.009ex;" alt="{\displaystyle \langle x,u_{\varphi }\rangle =\varphi (x)}"> for all x ∈ H. The inner product on the dual space H* satisfies <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \varphi ,\psi \rangle =\langle u_{\psi },u_{\varphi }\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>φ</mi> <mo>,</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \varphi ,\psi \rangle =\langle u_{\psi },u_{\varphi }\rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe81611f66827f550f48ab0e8d55854e834f8408" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.121ex; height:3.009ex;" alt="{\displaystyle \langle \varphi ,\psi \rangle =\langle u_{\psi },u_{\varphi }\rangle \,.}"> The reversal of order on the right-hand side restores linearity in φ from the antilinearity of uφ. In the real case, the antilinear isomorphism from H to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals. The representing vector uφ is obtained in the following way. When φ ≠ 0, the <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> F = Ker(φ) is a closed vector subspace of H, not equal to H, hence there exists a nonzero vector v orthogonal to F. The vector u is a suitable scalar multiple λv of v. The requirement that φ(v) = ⟨v, u⟩ yields <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\langle v,v\rangle ^{-1}\,{\overline {\varphi (v)}}\,v\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo>,</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=\langle v,v\rangle ^{-1}\,{\overline {\varphi (v)}}\,v\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0940b7ef7fe6dfc275ea7ab1961924b9158a047c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.367ex; height:3.676ex;" alt="{\displaystyle u=\langle v,v\rangle ^{-1}\,{\overline {\varphi (v)}}\,v\,.}"> This correspondence φ ↔ u is exploited by the <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> popular in <a href="/wiki/Physics" title="Physics">physics</a>.<a href="#cite_note-78">[76]</a> It is common in physics to assume that the inner product, denoted by ⟨x|y⟩, is linear on the right, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x|y\rangle =\langle y,x\rangle \,.}"> <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> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x|y\rangle =\langle y,x\rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d436ad7973ee47f13d01890d26cc98612fae99c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.402ex; height:2.843ex;" alt="{\displaystyle \langle x|y\rangle =\langle y,x\rangle \,.}"> The result ⟨x|y⟩ can be seen as the action of the linear functional ⟨x| (the bra) on the vector |y⟩ (the ket). The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the <a href="/wiki/Banach_space" title="Banach space">topological dual</a> of any inner product space can be identified with its completion.<a href="#cite_note-79">[77]</a> An immediate consequence of the Riesz representation theorem is also that a Hilbert space H is <a href="/wiki/Reflexive_space" title="Reflexive space">reflexive</a>, meaning that the natural map from H into its <a href="/wiki/Dual_space" title="Dual space">double dual space</a> is an isomorphism. <div class="mw-heading mw-heading3"><h3 id="Weakly-convergent_sequences">Weakly-convergent sequences</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=25" title="Edit section: Weakly-convergent sequences">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Weak_convergence_(Hilbert_space)" title="Weak convergence (Hilbert space)">Weak convergence (Hilbert space)</a></div> In a Hilbert space H, a sequence {xn} is <a href="/wiki/Weak_topology#Weak_convergence" title="Weak topology">weakly convergent</a> to a vector x ∈ H when <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n}\langle x_{n},v\rangle =\langle x,v\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n}\langle x_{n},v\rangle =\langle x,v\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc10be6704806b8ad1d07b418c54115e93863c0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.148ex; height:3.843ex;" alt="{\displaystyle \lim _{n}\langle x_{n},v\rangle =\langle x,v\rangle }"> for every v ∈ H. For example, any orthonormal sequence {fn} converges weakly to 0, as a consequence of <a href="#Bessel's_inequality">Bessel's inequality</a>. Every weakly convergent sequence {xn} is bounded, by the <a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">uniform boundedness principle</a>. Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (<a href="/wiki/Alaoglu%27s_theorem" class="mw-redirect" title="Alaoglu's theorem">Alaoglu's theorem</a>).<a href="#cite_note-80">[78]</a> This fact may be used to prove minimization results for continuous <a href="/wiki/Convex_function" title="Convex function">convex functionals</a>, in the same way that the <a href="/wiki/Bolzano%E2%80%93Weierstrass_theorem" title="Bolzano–Weierstrass theorem">Bolzano–Weierstrass theorem</a> is used for continuous functions on Rd. Among several variants, one simple statement is as follows:<a href="#cite_note-81">[79]</a> <dl><dd>If f : H → R is a convex continuous function such that f(x) tends to +∞ when ‖x‖ tends to ∞, then f admits a minimum at some point x0 ∈ H.</dd></dl> This fact (and its various generalizations) are fundamental for <a href="/wiki/Direct_method_in_the_calculus_of_variations" title="Direct method in the calculus of variations">direct methods</a> in the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space H are <a href="/wiki/Weak_topology" title="Weak topology">weakly compact</a>, since H is reflexive. The existence of weakly convergent subsequences is a special case of the <a href="/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian theorem</a>. <div class="mw-heading mw-heading3"><h3 id="Banach_space_properties">Banach space properties</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=26" title="Edit section: Banach space properties">edit</a>]</div> Any general property of <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a> continues to hold for Hilbert spaces. The <a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">open mapping theorem</a> states that a <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> linear transformation from one Banach space to another is an <a href="/wiki/Open_mapping" class="mw-redirect" title="Open mapping">open mapping</a> meaning that it sends open sets to open sets. A corollary is the <a href="/wiki/Bounded_inverse_theorem" class="mw-redirect" title="Bounded inverse theorem">bounded inverse theorem</a>, that a continuous and <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces.<a href="#cite_note-82">[80]</a> The open mapping theorem is equivalent to the <a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">closed graph theorem</a>, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a <a href="/wiki/Closed_set" title="Closed set">closed set</a>.<a href="#cite_note-83">[81]</a> In the case of Hilbert spaces, this is basic in the study of <a href="/wiki/Unbounded_operator" title="Unbounded operator">unbounded operators</a> (see <a href="/wiki/Closed_operator" class="mw-redirect" title="Closed operator">closed operator</a>). The (geometrical) <a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a> asserts that a closed convex set can be separated from any point outside it by means of a <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> of the Hilbert space. This is an immediate consequence of the <a href="#Best_approximation">best approximation</a> property: if y is the element of a closed convex set F closest to x, then the separating hyperplane is the plane perpendicular to the segment xy passing through its midpoint.<a href="#cite_note-84">[82]</a> <div class="mw-heading mw-heading2"><h2 id="Operators_on_Hilbert_spaces">Operators on Hilbert spaces</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=27" title="Edit section: Operators on Hilbert spaces">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Bounded_operators">Bounded operators</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=28" title="Edit section: Bounded operators">edit</a>]</div> The <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> A : H1 → H2 from a Hilbert space H1 to a second Hilbert space H2 are <a href="/wiki/Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded</a> in the sense that they map <a href="/wiki/Bounded_set" title="Bounded set">bounded sets</a> to bounded sets.<a href="#cite_note-85">[83]</a> Conversely, if an operator is bounded, then it is continuous. The space of such <a href="/wiki/Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded linear operators</a> has a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, the <a href="/wiki/Operator_norm" title="Operator norm">operator norm</a> given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lVert A\rVert =\sup {\bigl \{}\|Ax\|\mathrel {\big |} \|x\|\leq 1{\bigr \}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert A\rVert =\sup {\bigl \{}\|Ax\|\mathrel {\big |} \|x\|\leq 1{\bigr \}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa5bda3c3b0a00fe3f8551b4e4628c4d6288e89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.05ex; height:3.176ex;" alt="{\displaystyle \lVert A\rVert =\sup {\bigl \{}\|Ax\|\mathrel {\big |} \|x\|\leq 1{\bigr \}}\,.}"> The sum and the composite of two bounded linear operators is again bounded and linear. For y in H2, the map that sends x ∈ H1 to ⟨Ax, y⟩ is linear and continuous, and according to the <a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a> can therefore be represented in the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle x,A^{*}y\right\rangle =\langle Ax,y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mo>,</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>y</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle x,A^{*}y\right\rangle =\langle Ax,y\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b29adafb3f47953e72bb01aad30ad286e8ce98eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.296ex; height:2.843ex;" alt="{\displaystyle \left\langle x,A^{*}y\right\rangle =\langle Ax,y\rangle }"> for some vector A*y in H1. This defines another bounded linear operator A* : H2 → H1, the <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">adjoint</a> of A. The adjoint satisfies A** = A. When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of A can be shown to be <a href="/wiki/Riesz_representation_theorem#Adjoints_and_transposes" title="Riesz representation theorem">identical to</a> the <a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">transpose</a> tA : H2* → H1* of A, which by definition sends <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \in H_{2}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \in H_{2}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b567856dbcc5edd986c8a0354ec55b70722ecb8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.512ex; height:2.843ex;" alt="{\displaystyle \psi \in H_{2}^{*}}"> to the functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \circ A\in H_{1}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>∘</mo> <mi>A</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \circ A\in H_{1}^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebde178b2785096195c865497a755ef7b12841e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.096ex; height:2.843ex;" alt="{\displaystyle \psi \circ A\in H_{1}^{*}.}"> The set B(H) of all bounded linear operators on H (meaning operators H → H), together with the addition and composition operations, the norm and the adjoint operation, is a <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a>, which is a type of <a href="/wiki/Operator_algebra" title="Operator algebra">operator algebra</a>. An element A of B(H) is called 'self-adjoint' or 'Hermitian' if A* = A. If A is Hermitian and ⟨Ax, x⟩ ≥ 0 for every x, then A is called 'nonnegative', written A ≥ 0; if equality holds only when x = 0, then A is called 'positive'. The set of self adjoint operators admits a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>, in which A ≥ B if A − B ≥ 0. If A has the form B*B for some B, then A is nonnegative; if B is invertible, then A is positive. A converse is also true in the sense that, for a non-negative operator A, there exists a unique non-negative <a href="/wiki/Square_root_of_a_matrix" title="Square root of a matrix">square root</a> B such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B^{2}=B^{*}B\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>B</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B^{2}=B^{*}B\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ca56120d48281eac1949aca5c4a1d18d12d1a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.374ex; height:2.676ex;" alt="{\displaystyle A=B^{2}=B^{*}B\,.}"> In a sense made precise by the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a>, self-adjoint operators can usefully be thought of as operators that are "real". An element A of B(H) is called normal if A*A = AA*. Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operator <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {A+A^{*}}{2}}+i{\frac {A-A^{*}}{2i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>+</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>−</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {A+A^{*}}{2}}+i{\frac {A-A^{*}}{2i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b7b43f751581832f492a76d21d90aceb87865d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.918ex; height:5.343ex;" alt="{\displaystyle A={\frac {A+A^{*}}{2}}+i{\frac {A-A^{*}}{2i}}}"> that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts. An element U of B(H) is called <a href="/wiki/Unitary_operator" title="Unitary operator">unitary</a> if U is invertible and its inverse is given by U*. This can also be expressed by requiring that U be onto and ⟨Ux, Uy⟩ = ⟨x, y⟩ for all x, y ∈ H. The unitary operators form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> under composition, which is the <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a> of H. An element of B(H) is <a href="/wiki/Compact_operator" title="Compact operator">compact</a> if it sends bounded sets to <a href="/wiki/Relatively_compact" class="mw-redirect" title="Relatively compact">relatively compact</a> sets. Equivalently, a bounded operator T is compact if, for any bounded sequence {xk}, the sequence {Txk} has a convergent subsequence. Many <a href="/wiki/Integral_operator" title="Integral operator">integral operators</a> are compact, and in fact define a special class of operators known as <a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt operators</a> that are especially important in the study of <a href="/wiki/Integral_equation" title="Integral equation">integral equations</a>. <a href="/wiki/Fredholm_operator" title="Fredholm operator">Fredholm operators</a> differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional <a href="/wiki/Kernel_(linear_operator)" class="mw-redirect" title="Kernel (linear operator)">kernel</a> and <a href="/wiki/Cokernel" title="Cokernel">cokernel</a>. The index of a Fredholm operator T is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {index} T=\dim \ker T-\dim \operatorname {coker} T\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>index</mi> <mo>⁡</mo> <mi>T</mi> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mi>ker</mi> <mo>⁡</mo> <mi>T</mi> <mo>−</mo> <mi>dim</mi> <mo>⁡</mo> <mi>coker</mi> <mo>⁡</mo> <mi>T</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {index} T=\dim \ker T-\dim \operatorname {coker} T\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77927cb900b809d2100c945d0c0ffcdb9cbf187f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:35.597ex; height:2.343ex;" alt="{\displaystyle \operatorname {index} T=\dim \ker T-\dim \operatorname {coker} T\,.}"> The index is <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> invariant, and plays a deep role in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> via the <a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>. <div class="mw-heading mw-heading3"><h3 id="Unbounded_operators">Unbounded operators</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=29" title="Edit section: Unbounded operators">edit</a>]</div> <a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded operators</a> are also tractable in Hilbert spaces, and have important applications to <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>.<a href="#cite_note-86">[84]</a> An unbounded operator T on a Hilbert space H is defined as a linear operator whose domain D(T) is a linear subspace of H. Often the domain D(T) is a dense subspace of H, in which case T is known as a <a href="/wiki/Densely_defined_operator" title="Densely defined operator">densely defined operator</a>. The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint unbounded operators</a> play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space L2(R) are:<a href="#cite_note-87">[85]</a> <ul><li>A suitable extension of the differential operator <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Af)(x)=-i{\frac {\mathrm {d} }{\mathrm {d} x}}f(x)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Af)(x)=-i{\frac {\mathrm {d} }{\mathrm {d} x}}f(x)\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f052b6830adce7017b7d7e60a8f964e77603395" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.589ex; height:5.509ex;" alt="{\displaystyle (Af)(x)=-i{\frac {\mathrm {d} }{\mathrm {d} x}}f(x)\,,}"> where i is the imaginary unit and f is a differentiable function of compact support.</li> <li>The multiplication-by-x operator: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Bf)(x)=xf(x)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Bf)(x)=xf(x)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7fecc1d7fe4103f485fa583d8ed01b2d61708a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.871ex; height:2.843ex;" alt="{\displaystyle (Bf)(x)=xf(x)\,.}"></li></ul> These correspond to the <a href="/wiki/Momentum" title="Momentum">momentum</a> and <a href="/wiki/Position_operator" title="Position operator">position</a> observables, respectively. Neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L2(R). <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=30" title="Edit section: Constructions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Direct_sums">Direct sums</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=31" title="Edit section: Direct sums">edit</a>]</div> Two Hilbert spaces H1 and H2 can be combined into another Hilbert space, called the <a href="/wiki/Direct_sum_of_modules#Direct_sum_of_Hilbert_spaces" title="Direct sum of modules">(orthogonal) direct sum</a>,<a href="#cite_note-88">[86]</a> and denoted <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}\oplus H_{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}\oplus H_{2}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfae1d22f024f459b7993cb0457aa8575b0292a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.845ex; height:2.509ex;" alt="{\displaystyle H_{1}\oplus H_{2}\,,}"> consisting of the set of all <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> (x1, x2) where xi ∈ Hi, i = 1, 2, and inner product defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl \langle }(x_{1},x_{2}),(y_{1},y_{2}){\bigr \rangle }_{H_{1}\oplus H_{2}}=\left\langle x_{1},y_{1}\right\rangle _{H_{1}}+\left\langle x_{2},y_{2}\right\rangle _{H_{2}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">⟨</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">⟩</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl \langle }(x_{1},x_{2}),(y_{1},y_{2}){\bigr \rangle }_{H_{1}\oplus H_{2}}=\left\langle x_{1},y_{1}\right\rangle _{H_{1}}+\left\langle x_{2},y_{2}\right\rangle _{H_{2}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93bc21003f3808137fb4c3f5ada597ecc18dfafc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:50.648ex; height:3.676ex;" alt="{\displaystyle {\bigl \langle }(x_{1},x_{2}),(y_{1},y_{2}){\bigr \rangle }_{H_{1}\oplus H_{2}}=\left\langle x_{1},y_{1}\right\rangle _{H_{1}}+\left\langle x_{2},y_{2}\right\rangle _{H_{2}}\,.}"> More generally, if Hi is a family of Hilbert spaces indexed by i ∈ I, then the direct sum of the Hi, denoted <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{i\in I}H_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigoplus _{i\in I}H_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f61cfad2e924b1c5a4cdfe63d5ec8d7a176ef7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.629ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{i\in I}H_{i}}"> consists of the set of all indexed families <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(x_{i}\in H_{i}\mid i\in I)\in \prod _{i\in I}H_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(x_{i}\in H_{i}\mid i\in I)\in \prod _{i\in I}H_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a199f9cfb236ef57ac07da4f97bd2cfe26de7cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.618ex; height:5.676ex;" alt="{\displaystyle x=(x_{i}\in H_{i}\mid i\in I)\in \prod _{i\in I}H_{i}}"> in the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of the Hi such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i\in I}\|x_{i}\|^{2}<\infty \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i\in I}\|x_{i}\|^{2}<\infty \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a15cfa56f27c36c0d8091110b30eebfdad5c051f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.707ex; height:5.676ex;" alt="{\displaystyle \sum _{i\in I}\|x_{i}\|^{2}<\infty \,.}"> The inner product is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =\sum _{i\in I}\left\langle x_{i},y_{i}\right\rangle _{H_{i}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =\sum _{i\in I}\left\langle x_{i},y_{i}\right\rangle _{H_{i}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4aacc8f267ce5405bcd8d77550ae773254dc247" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.337ex; height:5.676ex;" alt="{\displaystyle \langle x,y\rangle =\sum _{i\in I}\left\langle x_{i},y_{i}\right\rangle _{H_{i}}\,.}"> Each of the Hi is included as a closed subspace in the direct sum of all of the Hi. Moreover, the Hi are pairwise orthogonal. Conversely, if there is a system of closed subspaces, Vi, i ∈ I, in a Hilbert space H, that are pairwise orthogonal and whose union is dense in H, then H is canonically isomorphic to the direct sum of Vi. In this case, H is called the internal direct sum of the Vi. A direct sum (internal or external) is also equipped with a family of orthogonal projections Ei onto the ith direct summand Hi. These projections are bounded, self-adjoint, <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a> operators that satisfy the orthogonality condition <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}E_{j}=0,\quad i\neq j\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}E_{j}=0,\quad i\neq j\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc313041cf3e24642f4f53b752db11de86c99c99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.65ex; height:2.843ex;" alt="{\displaystyle E_{i}E_{j}=0,\quad i\neq j\,.}"> The <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> for <a href="/wiki/Compact_operator" title="Compact operator">compact</a> self-adjoint operators on a Hilbert space H states that H splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the <a href="/wiki/Fock_space" title="Fock space">Fock space</a> of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional <a href="/wiki/Degrees_of_freedom_(mechanics)" title="Degrees of freedom (mechanics)">degree of freedom</a> for the quantum mechanical system. In <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>, the <a href="/wiki/Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a> guarantees that any <a href="/wiki/Unitary_representation" title="Unitary representation">unitary representation</a> of a <a href="/wiki/Compact_group" title="Compact group">compact group</a> on a Hilbert space splits as the direct sum of finite-dimensional representations. <div class="mw-heading mw-heading3"><h3 id="Tensor_products">Tensor products</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=32" title="Edit section: Tensor products">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">Tensor product of Hilbert spaces</a></div> If x1, y1 ∊ H1 and x2, y2 ∊ H2, then one defines an inner product on the (ordinary) <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> as follows. On <a href="/wiki/Simple_tensor" class="mw-redirect" title="Simple tensor">simple tensors</a>, let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x_{1}\otimes x_{2},\,y_{1}\otimes y_{2}\rangle =\langle x_{1},y_{1}\rangle \,\langle x_{2},y_{2}\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x_{1}\otimes x_{2},\,y_{1}\otimes y_{2}\rangle =\langle x_{1},y_{1}\rangle \,\langle x_{2},y_{2}\rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b591ee3265bc86c65b2299c3212c29b4918619" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.427ex; height:2.843ex;" alt="{\displaystyle \langle x_{1}\otimes x_{2},\,y_{1}\otimes y_{2}\rangle =\langle x_{1},y_{1}\rangle \,\langle x_{2},y_{2}\rangle \,.}"> This formula then extends by <a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinearity</a> to an inner product on H1 ⊗ H2. The Hilbertian tensor product of H1 and H2, sometimes denoted by H1 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\otimes }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>⊗</mo> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\otimes }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769aaba98f277892ad4872fa3a3620550e96323d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.843ex;" alt="{\displaystyle {\widehat {\otimes }}}"> H2, is the Hilbert space obtained by completing H1 ⊗ H2 for the metric associated to this inner product.<a href="#cite_note-89">[87]</a> An example is provided by the Hilbert space L2([0, 1]). The Hilbertian tensor product of two copies of L2([0, 1]) is isometrically and linearly isomorphic to the space L2([0, 1]2) of square-integrable functions on the square [0, 1]2. This isomorphism sends a simple tensor f1 ⊗ f2 to the function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s,t)\mapsto f_{1}(s)\,f_{2}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s,t)\mapsto f_{1}(s)\,f_{2}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bba73ee31532e31ff40fa8a8bb36865c597bb23c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.71ex; height:2.843ex;" alt="{\displaystyle (s,t)\mapsto f_{1}(s)\,f_{2}(t)}"> on the square. This example is typical in the following sense.<a href="#cite_note-90">[88]</a> Associated to every simple tensor product x1 ⊗ x2 is the rank one operator from H∗ 1 to H2 that maps a given x* ∈ H∗ 1 as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}\mapsto x^{*}(x_{1})x_{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{*}\mapsto x^{*}(x_{1})x_{2}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa601a323516b585fb98c5771bf04b90ee16f8e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.993ex; height:2.843ex;" alt="{\displaystyle x^{*}\mapsto x^{*}(x_{1})x_{2}\,.}"> This mapping defined on simple tensors extends to a linear identification between H1 ⊗ H2 and the space of finite rank operators from H∗ 1 to H2. This extends to a linear isometry of the Hilbertian tensor product H1 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\otimes }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>⊗</mo> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\otimes }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769aaba98f277892ad4872fa3a3620550e96323d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.843ex;" alt="{\displaystyle {\widehat {\otimes }}}"> H2 with the Hilbert space HS(H∗ 1, H2) of <a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt operators</a> from H∗ 1 to H2. <div class="mw-heading mw-heading2"><h2 id="Orthonormal_bases">Orthonormal bases</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=33" title="Edit section: Orthonormal bases">edit</a>]</div> The notion of an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> from linear algebra generalizes over to the case of Hilbert spaces.<a href="#cite_note-91">[89]</a> In a Hilbert space H, an orthonormal basis is a family {ek}k ∈ B of elements of H satisfying the conditions: <ol><li>Orthogonality: Every two different elements of B are orthogonal: ⟨ek, ej⟩ = 0 for all k, j ∈ B with k ≠ j.</li> <li>Normalization: Every element of the family has norm 1: ‖ek‖ = 1 for all k ∈ B.</li> <li>Completeness: The <a href="/wiki/Linear_span" title="Linear span">linear span</a> of the family ek, k ∈ B, is <a href="/wiki/Dense_set" title="Dense set">dense</a> in H.</li></ol> A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if B is <a href="/wiki/Countable_set" title="Countable set">countable</a>). Such a system is always <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a>. Despite the name, an orthonormal basis is not, in general, a basis in the sense of linear algebra (<a href="/wiki/Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel basis</a>). More precisely, an orthonormal basis is a Hamel basis if and only if the Hilbert space is a finite-dimensional vector space.<a href="#cite_note-92">[90]</a> Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as: <dl><dd>for every v ∈ H, if ⟨v, ek⟩ = 0 for all k ∈ B, then v = 0.</dd></dl> This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if S is any orthonormal set and v is orthogonal to S, then v is orthogonal to the closure of the linear span of S, which is the whole space. Examples of orthonormal bases include: <ul><li>the set {(1, 0, 0), (0, 1, 0), (0, 0, 1)} forms an orthonormal basis of R3 with the <a href="/wiki/Dot_product" title="Dot product">dot product</a>;</li> <li>the sequence { fn | n ∈ Z} with fn(x) = <a href="/wiki/Exponential_function" title="Exponential function">exp</a>(2πinx) forms an orthonormal basis of the complex space L2([0, 1]);</li></ul> In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>; to distinguish the two, the latter basis is also called a <a href="/wiki/Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel basis</a>. That the span of the basis vectors is dense implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique. <div class="mw-heading mw-heading3"><h3 id="Sequence_spaces">Sequence spaces</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=34" title="Edit section: Sequence spaces">edit</a>]</div> The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a4571ee9be10bd3c9df2480ab3d280f99e801a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{2}}"> of square-summable sequences of complex numbers is the set of infinite sequences<a href="#cite_note-Stein_2005-10">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{1},c_{2},c_{3},\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{1},c_{2},c_{3},\dots )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768f67354e8a3d14402b25c6570e502c4ae57d3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.818ex; height:2.843ex;" alt="{\displaystyle (c_{1},c_{2},c_{3},\dots )}"> of real or complex numbers such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}+\left|c_{3}\right|^{2}+\cdots <\infty \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>|</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>|</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}+\left|c_{3}\right|^{2}+\cdots <\infty \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34d0333aabdeb5ecdd32dff6435c0b21649646b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.928ex; height:3.343ex;" alt="{\displaystyle \left|c_{1}\right|^{2}+\left|c_{2}\right|^{2}+\left|c_{3}\right|^{2}+\cdots <\infty \,.}"> This space has an orthonormal basis: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e_{1}&=(1,0,0,\dots )\\e_{2}&=(0,1,0,\dots )\\&\ \ \vdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mo>⋮</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e_{1}&=(1,0,0,\dots )\\e_{2}&=(0,1,0,\dots )\\&\ \ \vdots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651d004b8d244263bc4b40ee99daeeaea3c47b83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:17.109ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}e_{1}&=(1,0,0,\dots )\\e_{2}&=(0,1,0,\dots )\\&\ \ \vdots \end{aligned}}}"> This space is the infinite-dimensional generalization of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{2}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{2}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac3233a9c36d0e75e9bfc58aab52554a80bbafd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.188ex; height:2.843ex;" alt="{\displaystyle \ell _{2}^{n}}"> space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that is <a href="/wiki/Closed_set" title="Closed set">closed</a> and <a href="/wiki/Bounded_set" title="Bounded set">bounded</a> is not necessarily <a href="/wiki/Sequentially_compact_space" title="Sequentially compact space">(sequentially) compact</a> (as is the case in all finite dimensional spaces). Indeed, the set of orthonormal vectors above shows this: It is an infinite sequence of vectors in the unit ball (i.e., the ball of points with norm less than or equal one). This set is clearly bounded and closed; yet, no subsequence of these vectors converges to anything and consequently the unit ball in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a4571ee9be10bd3c9df2480ab3d280f99e801a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{2}}"> is not compact. Intuitively, this is because "there is always another coordinate direction" into which the next elements of the sequence can evade. One can generalize the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a4571ee9be10bd3c9df2480ab3d280f99e801a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{2}}"> in many ways. For example, if B is any set, then one can form a Hilbert space of sequences with index set B, defined by<a href="#cite_note-93">[91]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(B)={\biggl \{}x:B\xrightarrow {x} \mathbb {C} \mathrel {\bigg |} \sum _{b\in B}\left|x(b)\right|^{2}<\infty {\biggr \}}\,.}"> <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>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mi>x</mi> <mo>:</mo> <mi>B</mi> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>x</mi> </mpadded> </mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>b</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 class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(B)={\biggl \{}x:B\xrightarrow {x} \mathbb {C} \mathrel {\bigg |} \sum _{b\in B}\left|x(b)\right|^{2}<\infty {\biggr \}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d626401d25c62bd23b60f8c6947ed391898efbd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.125ex; height:6.843ex;" alt="{\displaystyle \ell ^{2}(B)={\biggl \{}x:B\xrightarrow {x} \mathbb {C} \mathrel {\bigg |} \sum _{b\in B}\left|x(b)\right|^{2}<\infty {\biggr \}}\,.}"> The summation over B is here defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{b\in B}\left|x(b)\right|^{2}=\sup \sum _{n=1}^{N}\left|x(b_{n})\right|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</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> <msup> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{b\in B}\left|x(b)\right|^{2}=\sup \sum _{n=1}^{N}\left|x(b_{n})\right|^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923ebbaa57863084429d6ae2e1a09b598d76cc9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.658ex; height:7.509ex;" alt="{\displaystyle \sum _{b\in B}\left|x(b)\right|^{2}=\sup \sum _{n=1}^{N}\left|x(b_{n})\right|^{2}}"> the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> being taken over all finite subsets of B. It follows that, for this sum to be finite, every element of l2(B) has only countably many nonzero terms. This space becomes a Hilbert space with the inner product <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =\sum _{b\in B}x(b){\overline {y(b)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> <mi>x</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =\sum _{b\in B}x(b){\overline {y(b)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceafc84777c484d7336b63ae19a8373e061443ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.383ex; height:6.009ex;" alt="{\displaystyle \langle x,y\rangle =\sum _{b\in B}x(b){\overline {y(b)}}}"> for all x, y ∈ l2(B). Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality. An orthonormal basis of l2(B) is indexed by the set B, given by <dl><dd><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{b}(b')={\begin{cases}1&{\text{if }}b=b'\\0&{\text{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>b</mi> <mo>=</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{b}(b')={\begin{cases}1&{\text{if }}b=b'\\0&{\text{otherwise.}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b89bb9d648728d42a2333751eac8a71700c31d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.815ex; height:6.176ex;" alt="{\displaystyle e_{b}(b')={\begin{cases}1&{\text{if }}b=b'\\0&{\text{otherwise.}}\end{cases}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bessel's_inequality_and_Parseval's_formula">Bessel's inequality and Parseval's formula</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=35" title="Edit section: Bessel's inequality and Parseval's formula">edit</a>]</div> Let f1, …, fn be a finite orthonormal system in H. For an arbitrary vector x ∈ H, let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\sum _{j=1}^{n}\langle x,f_{j}\rangle \,f_{j}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\sum _{j=1}^{n}\langle x,f_{j}\rangle \,f_{j}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cae76e679dee41086182349e23adc6c0c983d60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.301ex; height:7.176ex;" alt="{\displaystyle y=\sum _{j=1}^{n}\langle x,f_{j}\rangle \,f_{j}\,.}"> Then ⟨x, fk⟩ = ⟨y, fk⟩ for every k = 1, …, n. It follows that x − y is orthogonal to each fk, hence x − y is orthogonal to y. Using the Pythagorean identity twice, it follows that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|^{2}=\|x-y\|^{2}+\|y\|^{2}\geq \|y\|^{2}=\sum _{j=1}^{n}{\bigl |}\langle x,f_{j}\rangle {\bigr |}^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|^{2}=\|x-y\|^{2}+\|y\|^{2}\geq \|y\|^{2}=\sum _{j=1}^{n}{\bigl |}\langle x,f_{j}\rangle {\bigr |}^{2}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6914b2f27fcd799379d1324efd6d26a92796afc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.964ex; height:7.176ex;" alt="{\displaystyle \|x\|^{2}=\|x-y\|^{2}+\|y\|^{2}\geq \|y\|^{2}=\sum _{j=1}^{n}{\bigl |}\langle x,f_{j}\rangle {\bigr |}^{2}\,.}"> Let {fi}, i ∈ I, be an arbitrary orthonormal system in H. Applying the preceding inequality to every finite subset J of I gives Bessel's inequality:<a href="#cite_note-94">[92]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i\in I}{\bigl |}\langle x,f_{i}\rangle {\bigr |}^{2}\leq \|x\|^{2},\quad x\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i\in I}{\bigl |}\langle x,f_{i}\rangle {\bigr |}^{2}\leq \|x\|^{2},\quad x\in H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04cd7cbd400cf024363716e694f90eabfe30a8d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.6ex; height:5.843ex;" alt="{\displaystyle \sum _{i\in I}{\bigl |}\langle x,f_{i}\rangle {\bigr |}^{2}\leq \|x\|^{2},\quad x\in H}"> (according to the definition of the <a href="/wiki/Series_(mathematics)#Summations_over_arbitrary_index_sets" title="Series (mathematics)">sum of an arbitrary family</a> of non-negative real numbers). Geometrically, Bessel's inequality implies that the orthogonal projection of x onto the linear subspace spanned by the fi has norm that does not exceed that of x. In two dimensions, this is the assertion that the length of the leg of a right triangle may not exceed the length of the hypotenuse. Bessel's inequality is a stepping stone to the stronger result called <a href="/wiki/Parseval_identity" class="mw-redirect" title="Parseval identity">Parseval's identity</a>, which governs the case when Bessel's inequality is actually an equality. By definition, if {ek}k ∈ B is an orthonormal basis of H, then every element x of H may be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\sum _{k\in B}\left\langle x,e_{k}\right\rangle \,e_{k}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mspace width="thinmathspace" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\sum _{k\in B}\left\langle x,e_{k}\right\rangle \,e_{k}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2998c111fdc5c46c1258e3ec87214b6b8961caef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.496ex; height:5.676ex;" alt="{\displaystyle x=\sum _{k\in B}\left\langle x,e_{k}\right\rangle \,e_{k}\,.}"> Even if B is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of countably many nonzero terms. This sum is called the Fourier expansion of x, and the individual coefficients ⟨x, ek⟩ are the Fourier coefficients of x. Parseval's identity then asserts that<a href="#cite_note-Hewitt_1965-95">[93]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|^{2}=\sum _{k\in B}|\langle x,e_{k}\rangle |^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <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> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|^{2}=\sum _{k\in B}|\langle x,e_{k}\rangle |^{2}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f86931997645328aefd2451525b8a4b3faf6d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.276ex; height:5.676ex;" alt="{\displaystyle \|x\|^{2}=\sum _{k\in B}|\langle x,e_{k}\rangle |^{2}\,.}"> Conversely,<a href="#cite_note-Hewitt_1965-95">[93]</a> if {ek} is an orthonormal set such that Parseval's identity holds for every x, then {ek} is an orthonormal basis. <div class="mw-heading mw-heading3"><h3 id="Hilbert_dimension">Hilbert dimension</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=36" title="Edit section: Hilbert dimension">edit</a>]</div> As a consequence of <a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a>, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same <a href="/wiki/Cardinal_number" title="Cardinal number">cardinality</a>, called the Hilbert dimension of the space.<a href="#cite_note-96">[94]</a> For instance, since l2(B) has an orthonormal basis indexed by B, its Hilbert dimension is the cardinality of B (which may be a finite integer, or a countable or uncountable <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a>). The Hilbert dimension is not greater than the <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">Hamel dimension</a> (the usual dimension of a vector space). The two dimensions are equal if and only if one of them is finite. As a consequence of Parseval's identity,<a href="#cite_note-97">[95]</a> if {ek}k ∈ B is an orthonormal basis of H, then the map Φ : H → l2(B) defined by Φ(x) = ⟨x, ek⟩k∈B is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl \langle }\Phi (x),\Phi (y){\bigr \rangle }_{l^{2}(B)}=\left\langle x,y\right\rangle _{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">⟨</mo> </mrow> </mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">⟩</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl \langle }\Phi (x),\Phi (y){\bigr \rangle }_{l^{2}(B)}=\left\langle x,y\right\rangle _{H}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e124d3427359d497ec8ec6795f810e9692257ab9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:26.888ex; height:3.676ex;" alt="{\displaystyle {\bigl \langle }\Phi (x),\Phi (y){\bigr \rangle }_{l^{2}(B)}=\left\langle x,y\right\rangle _{H}}"> for all x, y ∈ H. The <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a> of B is the Hilbert dimension of H. Thus every Hilbert space is isometrically isomorphic to a sequence space l2(B) for some set B. <div class="mw-heading mw-heading3"><h3 id="Separable_spaces">Separable spaces</h3>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=37" title="Edit section: Separable spaces">edit</a>]</div> By definition, a Hilbert space is <a href="/wiki/Separable_space" title="Separable space">separable</a> provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits a <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to the <a href="/wiki/Sequence_space#ℓp_spaces" title="Sequence space">square-summable sequence space</a> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e5f61f7f39a2ef38b3466db6274c425e782404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.671ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}.}"> In the past, Hilbert spaces were often required to be separable as part of the definition.<a href="#cite_note-98">[96]</a> <div class="mw-heading mw-heading4"><h4 id="In_quantum_field_theory">In quantum field theory</h4>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=38" title="Edit section: In quantum field theory">edit</a>]</div> Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space".<a href="#cite_note-99">[97]</a> Even in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, most of the Hilbert spaces are in fact separable, as stipulated by the <a href="/wiki/Wightman_axioms" title="Wightman axioms">Wightman axioms</a>. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of <a href="/wiki/Degrees_of_freedom_(mechanics)" title="Degrees of freedom (mechanics)">degrees of freedom</a> and any infinite <a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">Hilbert tensor product</a> (of spaces of dimension greater than one) is non-separable.<a href="#cite_note-Streater-100">[98]</a> For instance, a <a href="/wiki/Bosonic_field" title="Bosonic field">bosonic field</a> can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space.<a href="#cite_note-Streater-100">[98]</a> However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined). Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.<a href="#cite_note-Streater-100">[98]</a> <div class="mw-heading mw-heading2"><h2 id="Orthogonal_complements_and_projections">Orthogonal complements and projections</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=39" title="Edit section: Orthogonal complements and projections">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a></div> If S is a subset of a Hilbert space H, the set of vectors orthogonal to S is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\perp }=\left\{x\in H\mid \langle x,s\rangle =0\ {\text{ for all }}s\in S\right\}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>H</mi> <mo>∣</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{\perp }=\left\{x\in H\mid \langle x,s\rangle =0\ {\text{ for all }}s\in S\right\}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50d39d82dd2b8976b675f2799be00209c978c0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.567ex; height:3.176ex;" alt="{\displaystyle S^{\perp }=\left\{x\in H\mid \langle x,s\rangle =0\ {\text{ for all }}s\in S\right\}\,.}"> The set S⊥ is a <a href="/wiki/Closed_set" title="Closed set">closed</a> subspace of H (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. If V is a closed subspace of H, then V⊥ is called the <a href="/wiki/Orthogonal_complement" title="Orthogonal complement">orthogonal complement</a> of V. In fact, every x ∈ H can then be written uniquely as x = v + w, with v ∈ V and w ∈ V⊥. Therefore, H is the internal Hilbert direct sum of V and V⊥. The linear operator PV : H → H that maps x to v is called the <a href="/wiki/Orthogonal_projection" class="mw-redirect" title="Orthogonal projection">orthogonal projection</a> onto V. There is a <a href="/wiki/Natural_transformation" title="Natural transformation">natural</a> one-to-one correspondence between the set of all closed subspaces of H and the set of all bounded self-adjoint operators P such that P2 = P. Specifically, <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> Theorem — The orthogonal projection PV is a self-adjoint linear operator on H of norm ≤ 1 with the property P2 V = PV. Moreover, any self-adjoint linear operator E such that E2 = E is of the form PV, where V is the range of E. For every x in H, PV(x) is the unique element v of V that <a href="/wiki/Hilbert_projection_theorem" title="Hilbert projection theorem">minimizes the distance</a> ‖x − v‖. </div> This provides the geometrical interpretation of PV(x): it is the best approximation to x by elements of V.<a href="#cite_note-101">[99]</a> Projections PU and PV are called mutually orthogonal if PUPV = 0. This is equivalent to U and V being orthogonal as subspaces of H. The sum of the two projections PU and PV is a projection only if U and V are orthogonal to each other, and in that case PU + PV = PU+V.<a href="#cite_note-102">[100]</a> The composite PUPV is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that case PUPV = PU∩V.<a href="#cite_note-103">[101]</a> By restricting the codomain to the Hilbert space V, the orthogonal projection PV gives rise to a projection mapping π : H → V; it is the adjoint of the <a href="/wiki/Inclusion_mapping" class="mw-redirect" title="Inclusion mapping">inclusion mapping</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:V\to H\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>H</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:V\to H\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff8dd80c9bda997967981b74b00c080d6d9cb81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.238ex; height:2.509ex;" alt="{\displaystyle i:V\to H\,,}"> meaning that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle ix,y\right\rangle _{H}=\left\langle x,\pi y\right\rangle _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>⟨</mo> <mrow> <mi>i</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>π</mi> <mi>y</mi> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle ix,y\right\rangle _{H}=\left\langle x,\pi y\right\rangle _{V}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dedd63bfa32beb3519ab13b1feffd305e84c4b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.077ex; height:3.009ex;" alt="{\displaystyle \left\langle ix,y\right\rangle _{H}=\left\langle x,\pi y\right\rangle _{V}}"> for all x ∈ V and y ∈ H. The operator norm of the orthogonal projection PV onto a nonzero closed subspace V is equal to 1: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|P_{V}\|=\sup _{x\in H,x\neq 0}{\frac {\|P_{V}x\|}{\|x\|}}=1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>H</mi> <mo>,</mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|P_{V}\|=\sup _{x\in H,x\neq 0}{\frac {\|P_{V}x\|}{\|x\|}}=1\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a34fe5aa961ed70c750598d753f451aae687d3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.566ex; height:6.676ex;" alt="{\displaystyle \|P_{V}\|=\sup _{x\in H,x\neq 0}{\frac {\|P_{V}x\|}{\|x\|}}=1\,.}"> Every closed subspace V of a Hilbert space is therefore the image of an operator P of norm one such that P2 = P. The property of possessing appropriate projection operators characterizes Hilbert spaces:<a href="#cite_note-104">[102]</a> <ul><li>A Banach space of dimension higher than 2 is (isometrically) a Hilbert space if and only if, for every closed subspace V, there is an operator PV of norm one whose image is V such that P2 V = PV.</li></ul> While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> can itself be characterized in terms of the presence of complementary subspaces:<a href="#cite_note-105">[103]</a> <ul><li>A Banach space X is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspace V, there is a closed subspace W such that X is equal to the internal direct sum V ⊕ W.</li></ul> The orthogonal complement satisfies some more elementary results. It is a <a href="/wiki/Monotone_function" class="mw-redirect" title="Monotone function">monotone function</a> in the sense that if U ⊂ V, then V⊥ ⊆ U⊥ with equality holding if and only if V is contained in the <a href="/wiki/Closure_(topology)" title="Closure (topology)">closure</a> of U. This result is a special case of the <a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a>. The closure of a subspace can be completely characterized in terms of the orthogonal complement: if V is a subspace of H, then the closure of V is equal to V⊥⊥. The orthogonal complement is thus a <a href="/wiki/Galois_connection" title="Galois connection">Galois connection</a> on the <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements:<a href="#cite_note-106">[104]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\biggl (}\sum _{i}V_{i}{\biggr )}^{\perp }=\bigcap _{i}V_{i}^{\perp }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\biggl (}\sum _{i}V_{i}{\biggr )}^{\perp }=\bigcap _{i}V_{i}^{\perp }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9630ac35d7d98746b5461804660f51e13861fcf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.745ex; height:7.176ex;" alt="{\displaystyle {\biggl (}\sum _{i}V_{i}{\biggr )}^{\perp }=\bigcap _{i}V_{i}^{\perp }\,.}"> If the Vi are in addition closed, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\sum _{i}V_{i}^{\perp }{\vphantom {\Big |}}}}={\biggl (}\bigcap _{i}V_{i}{\biggr )}^{\perp }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">|</mo> </mrow> </mphantom> </mpadded> </mrow> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\sum _{i}V_{i}^{\perp }{\vphantom {\Big |}}}}={\biggl (}\bigcap _{i}V_{i}{\biggr )}^{\perp }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c7dc911875bb0fc30d91e4ee1c62f8ae5867f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.86ex; height:7.176ex;" alt="{\displaystyle {\overline {\sum _{i}V_{i}^{\perp }{\vphantom {\Big |}}}}={\biggl (}\bigcap _{i}V_{i}{\biggr )}^{\perp }\,.}"> <div class="mw-heading mw-heading2"><h2 id="Spectral_theory">Spectral theory</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=40" title="Edit section: Spectral theory">edit</a>]</div> There is a well-developed <a href="/wiki/Spectral_theory" title="Spectral theory">spectral theory</a> for self-adjoint operators in a Hilbert space, that is roughly analogous to the study of <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a> over the reals or self-adjoint matrices over the complex numbers.<a href="#cite_note-107">[105]</a> In the same sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of orthogonal projection operators. The <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectrum of an operator</a> T, denoted σ(T), is the set of complex numbers λ such that T − λ lacks a continuous inverse. If T is bounded, then the spectrum is always a <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact set</a> in the complex plane, and lies inside the disc |z| ≤ ‖T‖. If T is self-adjoint, then the spectrum is real. In fact, it is contained in the interval [m, M] where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=\inf _{\|x\|=1}\langle Tx,x\rangle \,,\quad M=\sup _{\|x\|=1}\langle Tx,x\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>M</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=\inf _{\|x\|=1}\langle Tx,x\rangle \,,\quad M=\sup _{\|x\|=1}\langle Tx,x\rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc60676b6b2cc94cd85e7447575d20dc96d68cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.104ex; height:5.009ex;" alt="{\displaystyle m=\inf _{\|x\|=1}\langle Tx,x\rangle \,,\quad M=\sup _{\|x\|=1}\langle Tx,x\rangle \,.}"> Moreover, m and M are both actually contained within the spectrum. The eigenspaces of an operator T are given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\lambda }=\ker(T-\lambda )\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>=</mo> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\lambda }=\ker(T-\lambda )\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2822ca54eb778655b2c52f2efa28394566abae91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.067ex; height:2.843ex;" alt="{\displaystyle H_{\lambda }=\ker(T-\lambda )\,.}"> Unlike with finite matrices, not every element of the spectrum of T must be an eigenvalue: the linear operator T − λ may only lack an inverse because it is not surjective. Elements of the spectrum of an operator in the general sense are known as spectral values. Since spectral values need not be eigenvalues, the spectral decomposition is often more subtle than in finite dimensions. However, the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> of a self-adjoint operator T takes a particularly simple form if, in addition, T is assumed to be a <a href="/wiki/Compact_operator" title="Compact operator">compact operator</a>. The <a href="/wiki/Compact_operator_on_Hilbert_space#Spectral_theorem" title="Compact operator on Hilbert space">spectral theorem for compact self-adjoint operators</a> states:<a href="#cite_note-108">[106]</a> <ul><li>A compact self-adjoint operator T has only countably (or finitely) many spectral values. The spectrum of T has no <a href="/wiki/Limit_point" class="mw-redirect" title="Limit point">limit point</a> in the complex plane except possibly zero. The eigenspaces of T decompose H into an orthogonal direct sum: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\bigoplus _{\lambda \in \sigma (T)}H_{\lambda }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </munder> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\bigoplus _{\lambda \in \sigma (T)}H_{\lambda }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70108e3beaada58a9e87760f340e7d87ecb0c55d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.136ex; height:6.009ex;" alt="{\displaystyle H=\bigoplus _{\lambda \in \sigma (T)}H_{\lambda }\,.}"> Moreover, if Eλ denotes the orthogonal projection onto the eigenspace Hλ, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\sum _{\lambda \in \sigma (T)}\lambda E_{\lambda }\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mi>λ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\sum _{\lambda \in \sigma (T)}\lambda E_{\lambda }\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e1f9f670a5b8f4d6debb1b646ebd41b2588570" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.848ex; height:6.009ex;" alt="{\displaystyle T=\sum _{\lambda \in \sigma (T)}\lambda E_{\lambda }\,,}"> where the sum converges with respect to the norm on B(H).</li></ul> This theorem plays a fundamental role in the theory of <a href="/wiki/Integral_equation" title="Integral equation">integral equations</a>, as many integral operators are compact, in particular those that arise from <a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt operators</a>. The general spectral theorem for self-adjoint operators involves a kind of operator-valued <a href="/wiki/Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a>, rather than an infinite summation.<a href="#cite_note-109">[107]</a> The spectral family associated to T associates to each real number λ an operator Eλ, which is the projection onto the nullspace of the operator (T − λ)+, where the positive part of a self-adjoint operator is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{+}={\tfrac {1}{2}}{\Bigl (}{\sqrt {A^{2}}}+A{\Bigr )}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{+}={\tfrac {1}{2}}{\Bigl (}{\sqrt {A^{2}}}+A{\Bigr )}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a99be7cf627f786503cca22294360ada0038c0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.525ex; height:4.843ex;" alt="{\displaystyle A^{+}={\tfrac {1}{2}}{\Bigl (}{\sqrt {A^{2}}}+A{\Bigr )}\,.}"> The operators Eλ are monotone increasing relative to the partial order defined on self-adjoint operators; the eigenvalues correspond precisely to the jump discontinuities. One has the spectral theorem, which asserts <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\int _{\mathbb {R} }\lambda \,\mathrm {d} E_{\lambda }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>λ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\int _{\mathbb {R} }\lambda \,\mathrm {d} E_{\lambda }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6337933ee817e34327fbb88c0f4fb5094f5794a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.808ex; height:5.676ex;" alt="{\displaystyle T=\int _{\mathbb {R} }\lambda \,\mathrm {d} E_{\lambda }\,.}"> The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm on B(H). In particular, one has the ordinary scalar-valued integral representation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Tx,y\rangle =\int _{\mathbb {R} }\lambda \,\mathrm {d} \langle E_{\lambda }x,y\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>λ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Tx,y\rangle =\int _{\mathbb {R} }\lambda \,\mathrm {d} \langle E_{\lambda }x,y\rangle \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266e2f6e28456de2af73cedddfd75bdc5f30c18a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.465ex; height:5.676ex;" alt="{\displaystyle \langle Tx,y\rangle =\int _{\mathbb {R} }\lambda \,\mathrm {d} \langle E_{\lambda }x,y\rangle \,.}"> A somewhat similar spectral decomposition holds for normal operators, although because the spectrum may now contain non-real complex numbers, the operator-valued Stieltjes measure dEλ must instead be replaced by a <a href="/wiki/Resolution_of_the_identity" class="mw-redirect" title="Resolution of the identity">resolution of the identity</a>. A major application of spectral methods is the <a href="/wiki/Spectral_mapping_theorem" class="mw-redirect" title="Spectral mapping theorem">spectral mapping theorem</a>, which allows one to apply to a self-adjoint operator T any continuous complex function f defined on the spectrum of T by forming the integral <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(T)=\int _{\sigma (T)}f(\lambda )\,\mathrm {d} E_{\lambda }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T)=\int _{\sigma (T)}f(\lambda )\,\mathrm {d} E_{\lambda }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36bf8ffa0897168c6173ebf83aafd9d1d7735435" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.174ex; height:6.009ex;" alt="{\displaystyle f(T)=\int _{\sigma (T)}f(\lambda )\,\mathrm {d} E_{\lambda }\,.}"> The resulting <a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous functional calculus</a> has applications in particular to <a href="/wiki/Pseudodifferential_operators" class="mw-redirect" title="Pseudodifferential operators">pseudodifferential operators</a>.<a href="#cite_note-110">[108]</a> The spectral theory of unbounded self-adjoint operators is only marginally more difficult than for bounded operators. The spectrum of an unbounded operator is defined in precisely the same way as for bounded operators: λ is a spectral value if the <a href="/wiki/Resolvent_operator" class="mw-redirect" title="Resolvent operator">resolvent operator</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\lambda }=(T-\lambda )^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\lambda }=(T-\lambda )^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ab87e56c41c0896676a628685b26e77c4dcebe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.027ex; height:3.176ex;" alt="{\displaystyle R_{\lambda }=(T-\lambda )^{-1}}"> fails to be a well-defined continuous operator. The self-adjointness of T still guarantees that the spectrum is real. Thus the essential idea of working with unbounded operators is to look instead at the resolvent Rλ where λ is nonreal. This is a bounded normal operator, which admits a spectral representation that can then be transferred to a spectral representation of T itself. A similar strategy is used, for instance, to study the spectrum of the Laplace operator: rather than address the operator directly, one instead looks as an associated resolvent such as a <a href="/wiki/Riesz_potential" title="Riesz potential">Riesz potential</a> or <a href="/wiki/Bessel_potential" title="Bessel potential">Bessel potential</a>. A precise version of the spectral theorem in this case is:<a href="#cite_note-111">[109]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> Theorem — Given a densely defined self-adjoint operator T on a Hilbert space H, there corresponds a unique <a href="/wiki/Resolution_of_the_identity" class="mw-redirect" title="Resolution of the identity">resolution of the identity</a> E on the Borel sets of R, such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Tx,y\rangle =\int _{\mathbb {R} }\lambda \,\mathrm {d} E_{x,y}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>λ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Tx,y\rangle =\int _{\mathbb {R} }\lambda \,\mathrm {d} E_{x,y}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43acc7d3fc1e95faa2b9d5b000516abdaa85f177" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.523ex; height:5.676ex;" alt="{\displaystyle \langle Tx,y\rangle =\int _{\mathbb {R} }\lambda \,\mathrm {d} E_{x,y}(\lambda )}"> for all x ∈ D(T) and y ∈ H. The spectral measure E is concentrated on the spectrum of T. </div> There is also a version of the spectral theorem that applies to unbounded normal operators. <div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=41" title="Edit section: In popular culture">edit</a>]</div> In <a href="/wiki/Gravity%27s_Rainbow" title="Gravity's Rainbow">Gravity's Rainbow</a> (1973), a novel by <a href="/wiki/Thomas_Pynchon" title="Thomas Pynchon">Thomas Pynchon</a>, one of the characters is called "Sammy Hilbert-Spaess", a pun on "Hilbert Space". The novel refers also to <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a>.<a href="#cite_note-112">[110]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=42" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 21em;"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach space</a> – Normed vector space that is complete</li> <li><a href="/wiki/Fock_space" title="Fock space">Fock space</a> – Multi particle state space</li> <li><a href="/wiki/Fundamental_theorem_of_Hilbert_spaces" title="Fundamental theorem of Hilbert spaces">Fundamental theorem of Hilbert spaces</a></li> <li><a href="/wiki/Hadamard_space" title="Hadamard space">Hadamard space</a> – geodesically complete metric space of non-positive curvaturePages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> – Type of topological space</li> <li><a href="/wiki/Hilbert_algebra_(disambiguation)" class="mw-redirect mw-disambig" title="Hilbert algebra (disambiguation)">Hilbert algebra</a></li> <li><a href="/wiki/Hilbert_C*-module" title="Hilbert C*-module">Hilbert C*-module</a> – Mathematical objects that generalise the notion of Hilbert spaces</li> <li><a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a> – Manifold modelled on Hilbert spaces</li> <li><a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a> – Generalization of inner products that applies to all normed spaces</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a> – A vector space with a topology defined by convex open sets</li> <li><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a> – Mathematical field of study</li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator topologies</a> – Topologies on the set of operators on a Hilbert space</li> <li><a href="/wiki/Quantum_state_space" title="Quantum state space">Quantum state space</a> – Mathematical space representing physical quantum systems</li> <li><a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">Rigged Hilbert space</a> – Construction linking the study of "bound" and continuous eigenvalues in functional analysis</li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector space</a> – Vector space with a notion of nearness</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Remarks">Remarks</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=43" title="Edit section: Remarks">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-5"><a href="#cite_ref-5">^</a> In some conventions, inner products are linear in their second arguments instead. </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> The eigenvalues of the Fredholm kernel are <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/λ⁠, which tend to zero. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=44" title="Edit section: Notes">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFAxler2014">Axler 2014</a>, p. 164 §6.2 </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> However, some sources call finite-dimensional spaces with these properties pre-Hilbert spaces, reserving the term "Hilbert space" for infinite-dimensional spaces; see, e.g., <a href="#CITEREFLevitan2001">Levitan 2001</a>. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFMarsden1974">Marsden 1974</a>, §2.8 </li> <li id="cite_note-General-4"><a href="#cite_ref-General_4-0">^</a> The mathematical material in this section can be found in any good textbook on functional analysis, such as <a href="#CITEREFDieudonné1960">Dieudonné (1960)</a>, <a href="#CITEREFHewittStromberg1965">Hewitt & Stromberg (1965)</a>, <a href="#CITEREFReedSimon1980">Reed & Simon (1980)</a> or <a href="#CITEREFRudin1987">Rudin (1987)</a>. </li> <li id="cite_note-FOOTNOTESchaeferWolff1999122–202-6"><a href="#cite_ref-FOOTNOTESchaeferWolff1999122–202_6-0">^</a> <a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, pp. 122–202. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFDieudonné1960">Dieudonné 1960</a>, §6.2 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFRoman2008">Roman 2008</a>, p. 327 </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFRoman2008">Roman 2008</a>, p. 330 Theorem 13.8 </li> <li id="cite_note-Stein_2005-10">^ <a href="#cite_ref-Stein_2005_10-0">a</a> <a href="#cite_ref-Stein_2005_10-1">b</a> <a href="#CITEREFSteinShakarchi2005">Stein & Shakarchi 2005</a>, p. 163 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFDieudonné1960">Dieudonné 1960</a> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Largely from the work of <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a>, at the urging of <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">August Ferdinand Möbius</a> (<a href="#CITEREFBoyerMerzbach1991">Boyer & Merzbach 1991</a>, pp. 584–586). The first modern axiomatic account of abstract vector spaces ultimately appeared in <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>'s 1888 account (<a href="#CITEREFGrattan-Guinness2000">Grattan-Guinness 2000</a>, §5.2.2; <a href="#CITEREFO'ConnorRobertson1996">O'Connor & Robertson 1996</a>). </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> A detailed account of the history of Hilbert spaces can be found in <a href="#CITEREFBourbaki1987">Bourbaki 1987</a>. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFSchmidt1908">Schmidt 1908</a> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFTitchmarsh1946">Titchmarsh 1946</a>, §IX.1 </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFLebesgue1904">Lebesgue 1904</a>. Further details on the history of integration theory can be found in <a href="#CITEREFBourbaki1987">Bourbaki (1987)</a> and <a href="#CITEREFSaks2005">Saks (2005)</a>. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFBourbaki1987">Bourbaki 1987</a>. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="#CITEREFDunfordSchwartz1958">Dunford & Schwartz 1958</a>, §IV.16 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> In <a href="#CITEREFDunfordSchwartz1958">Dunford & Schwartz (1958</a>, §IV.16), the result that every linear functional on L2[0,1] is represented by integration is jointly attributed to <a href="#CITEREFFréchet1907">Fréchet (1907)</a> and <a href="#CITEREFRiesz1907">Riesz (1907)</a>. The general result, that the dual of a Hilbert space is identified with the Hilbert space itself, can be found in <a href="#CITEREFRiesz1934">Riesz (1934)</a>. </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFvon_Neumann1929">von Neumann 1929</a>. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFKline1972">Kline 1972</a>, p. 1092 </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFHilbertNordheimvon_Neumann1927">Hilbert, Nordheim & von Neumann 1927</a> </li> <li id="cite_note-Weyl31-23">^ <a href="#cite_ref-Weyl31_23-0">a</a> <a href="#cite_ref-Weyl31_23-1">b</a> <a href="#CITEREFWeyl1931">Weyl 1931</a>. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFPrugovečki1981">Prugovečki 1981</a>, pp. 1–10. </li> <li id="cite_note-von_Neumann_1932-25">^ <a href="#cite_ref-von_Neumann_1932_25-0">a</a> <a href="#cite_ref-von_Neumann_1932_25-1">b</a> <a href="#CITEREFvon_Neumann1932">von Neumann 1932</a> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFPeres1993">Peres 1993</a>, pp. 79–99. </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFMurphy1990">Murphy 1990</a>, p. 112 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="#CITEREFMurphy1990">Murphy 1990</a>, p. 72 </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="#CITEREFHalmos1957">Halmos 1957</a>, Section 42. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <a href="#CITEREFHewittStromberg1965">Hewitt & Stromberg 1965</a>. </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="/wiki/Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_773.htm">"Chapter 22"</a>. <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61272-0" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+22&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=Washington+D.C.%3B+New+York&rft.series=Applied+Mathematics+Series&rft.pages=773&rft.edition=Ninth+reprint+with+additional+corrections+of+tenth+original+printing+with+corrections+%28December+1972%29%3B+first&rft.pub=United+States+Department+of+Commerce%2C+National+Bureau+of+Standards%3B+Dover+Publications&rft.date=1983&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0167642%23id-name%3DMR&rft_id=info%3Alccn%2F64-60036&rft.isbn=978-0-486-61272-0&rft_id=http%3A%2F%2Fwww.math.ubc.ca%2F~cbm%2Faands%2Fpage_773.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"> </li> <li id="cite_note-BeJoSc81-32">^ <a href="#cite_ref-BeJoSc81_32-0">a</a> <a href="#cite_ref-BeJoSc81_32-1">b</a> <a href="#CITEREFBersJohnSchechter1981">Bers, John & Schechter 1981</a>. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <a href="#CITEREFGiusti2003">Giusti 2003</a>. </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <a href="#CITEREFStein1970">Stein 1970</a> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> Details can be found in <a href="#CITEREFWarner1983">Warner (1983)</a>. </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> A general reference on Hardy spaces is the book <a href="#CITEREFDuren1970">Duren (1970)</a>. </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="#CITEREFKrantz2002">Krantz 2002</a>, §1.4 </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFKrantz2002">Krantz 2002</a>, §1.5 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFYoung1988">Young 1988</a>, Chapter 9. </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <a href="#CITEREFPedersen1995">Pedersen 1995</a>, §4.4 </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> More detail on finite element methods from this point of view can be found in <a href="#CITEREFBrennerScott2005">Brenner & Scott (2005)</a>. </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <a href="#CITEREFBrezis2010">Brezis 2010</a>, section 9.5 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <a href="#CITEREFEvans1998">Evans 1998</a> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <a href="#CITEREFPathria1996">Pathria (1996)</a>, Chapters 2 and 3 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <a href="#CITEREFEinsiedlerWard2011">Einsiedler & Ward (2011)</a>, Proposition 2.14. </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <a href="#CITEREFReedSimon1980">Reed & Simon 1980</a> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> A treatment of Fourier series from this point of view is available, for instance, in <a href="#CITEREFRudin1987">Rudin (1987)</a> or <a href="#CITEREFFolland2009">Folland (2009)</a>. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <a href="#CITEREFHalmos1957">Halmos 1957</a>, §5 </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <a href="#CITEREFBachmanNariciBeckenstein2000">Bachman, Narici & Beckenstein 2000</a> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>, §IV.2. </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <a href="#CITEREFLanczos1988">Lanczos 1988</a>, pp. 212–213 </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <a href="#CITEREFLanczos1988">Lanczos 1988</a>, Equation 4-3.10 </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> The classic reference for spectral methods is <a href="#CITEREFCourantHilbert1953">Courant & Hilbert 1953</a>. A more up-to-date account is <a href="#CITEREFReedSimon1975">Reed & Simon 1975</a>. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <a href="#CITEREFKac1966">Kac 1966</a> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <a href="#CITEREFvon_Neumann1955">von Neumann 1955</a> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="#CITEREFHolevo2001">Holevo 2001</a>, p. 17 </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <a href="#CITEREFRieffelPolak2011">Rieffel & Polak 2011</a>, p. 55 </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <a href="#CITEREFPeres1993">Peres 1993</a>, p. 101 </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <a href="#CITEREFPeres1993">Peres 1993</a>, pp. 73 </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <a href="#CITEREFNielsenChuang2000">Nielsen & Chuang 2000</a>, p. 90 </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <a href="#CITEREFBillingsley1986">Billingsley (1986)</a>, p. 477, ex. 34.13}} </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <a href="#CITEREFStapleton1995">Stapleton 1995</a> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <a href="#CITEREFHewittStromberg1965">Hewitt & Stromberg (1965)</a>, Exercise 16.45. </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <a href="#CITEREFKaratzasShreve2019">Karatzas & Shreve 2019</a>, Chapter 3 </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <a href="#CITEREFStroock2011">Stroock (2011)</a>, Chapter 8. </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHermann_Weyl2009" class="citation cs2"><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> (2009), "Mind and nature", Mind and nature: selected writings on philosophy, mathematics, and physics, Princeton University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mind+and+nature&rft.btitle=Mind+and+nature%3A+selected+writings+on+philosophy%2C+mathematics%2C+and+physics&rft.pub=Princeton+University+Press&rft.date=2009&rft.au=Hermann+Weyl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">. </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerthier2020" class="citation cs2">Berthier, M. (2020), "Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit", The Journal of Mathematical Neuroscience, 10 (1): 14, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1186%2Fs13408-020-00092-x">10.1186/s13408-020-00092-x</a>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7481323">7481323</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32902776">32902776</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Mathematical+Neuroscience&rft.atitle=Geometry+of+color+perception.+Part+2%3A+perceived+colors+from+real+quantum+states+and+Hering%27s+rebit&rft.volume=10&rft.issue=1&rft.pages=14&rft.date=2020&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7481323%23id-name%3DPMC&rft_id=info%3Apmid%2F32902776&rft_id=info%3Adoi%2F10.1186%2Fs13408-020-00092-x&rft.aulast=Berthier&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">. </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <a href="#CITEREFReedSimon1980">Reed & Simon 1980</a>, Theorem 12.6 </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <a href="#CITEREFReedSimon1980">Reed & Simon 1980</a>, p. 38 </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <a href="#CITEREFYoung1988">Young 1988</a>, p. 23. </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <a href="#CITEREFClarkson1936">Clarkson 1936</a>. </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <a href="#CITEREFRudin1987">Rudin 1987</a>, Theorem 4.10 </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <a href="#CITEREFDunfordSchwartz1958">Dunford & Schwartz 1958</a>, II.4.29 </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <a href="#CITEREFRudin1987">Rudin 1987</a>, Theorem 4.11 </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlanchetCharbit2014" class="citation book cs1">Blanchet, Gérard; Charbit, Maurice (2014). Digital Signal and Image Processing Using MATLAB. Vol. 1 (Second ed.). New Jersey: Wiley. pp. 349–360. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1848216402" title="Special:BookSources/978-1848216402"><bdi>978-1848216402</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Signal+and+Image+Processing+Using+MATLAB&rft.place=New+Jersey&rft.pages=349-360&rft.edition=Second&rft.pub=Wiley&rft.date=2014&rft.isbn=978-1848216402&rft.aulast=Blanchet&rft.aufirst=G%C3%A9rard&rft.au=Charbit%2C+Maurice&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"> </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <a href="#CITEREFWeidmann1980">Weidmann 1980</a>, Theorem 4.8 </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <a href="#CITEREFPeres1993">Peres 1993</a>, pp. 77–78. </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <a href="#CITEREFWeidmann1980">Weidmann (1980)</a>, Exercise 4.11. </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <a href="#CITEREFWeidmann1980">Weidmann 1980</a>, §4.5 </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <a href="#CITEREFButtazzoGiaquintaHildebrandt1998">Buttazzo, Giaquinta & Hildebrandt 1998</a>, Theorem 5.17 </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> <a href="#CITEREFHalmos1982">Halmos 1982</a>, Problem 52, 58 </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> <a href="#CITEREFRudin1973">Rudin 1973</a> </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> <a href="#CITEREFTrèves1967">Trèves 1967</a>, Chapter 18 </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> A general reference for this section is <a href="#CITEREFRudin1973">Rudin (1973)</a>, chapter 12. </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> See <a href="#CITEREFPrugovečki1981">Prugovečki (1981)</a>, <a href="#CITEREFReedSimon1980">Reed & Simon (1980</a>, Chapter VIII) and <a href="#CITEREFFolland1989">Folland (1989)</a>. </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> <a href="#CITEREFPrugovečki1981">Prugovečki 1981</a>, III, §1.4 </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> <a href="#CITEREFDunfordSchwartz1958">Dunford & Schwartz 1958</a>, IV.4.17-18 </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <a href="#CITEREFWeidmann1980">Weidmann 1980</a>, §3.4 </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <a href="#CITEREFKadisonRingrose1983">Kadison & Ringrose 1983</a>, Theorem 2.6.4 </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> <a href="#CITEREFDunfordSchwartz1958">Dunford & Schwartz 1958</a>, §IV.4. </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> <a href="#CITEREFRoman2008">Roman 2008</a>, p. 218 </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> <a href="#CITEREFRudin1987">Rudin 1987</a>, Definition 3.7 </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> For the case of finite index sets, see, for instance, <a href="#CITEREFHalmos1957">Halmos 1957</a>, §5. For infinite index sets, see <a href="#CITEREFWeidmann1980">Weidmann 1980</a>, Theorem 3.6. </li> <li id="cite_note-Hewitt_1965-95">^ <a href="#cite_ref-Hewitt_1965_95-0">a</a> <a href="#cite_ref-Hewitt_1965_95-1">b</a> <a href="#CITEREFHewittStromberg1965">Hewitt & Stromberg (1965)</a>, Theorem 16.26. </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> <a href="#CITEREFLevitan2001">Levitan 2001</a>. Many authors, such as <a href="#CITEREFDunfordSchwartz1958">Dunford & Schwartz (1958</a>, §IV.4), refer to this just as the dimension. Unless the Hilbert space is finite dimensional, this is not the same thing as its dimension as a linear space (the cardinality of a Hamel basis). </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> <a href="#CITEREFHewittStromberg1965">Hewitt & Stromberg (1965)</a>, Theorem 16.29. </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> <a href="#CITEREFPrugovečki1981">Prugovečki 1981</a>, I, §4.2 </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <a href="#CITEREFvon_Neumann1955">von Neumann (1955)</a> defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism with l2. The convention still persists in most rigorous treatments of quantum mechanics; see for instance <a href="#CITEREFSobrino1996">Sobrino 1996</a>, Appendix B. </li> <li id="cite_note-Streater-100">^ <a href="#cite_ref-Streater_100-0">a</a> <a href="#cite_ref-Streater_100-1">b</a> <a href="#cite_ref-Streater_100-2">c</a> <a href="#CITEREFStreaterWightman1964">Streater & Wightman 1964</a>, pp. 86–87 </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <a href="#CITEREFYoung1988">Young 1988</a>, Theorem 15.3 </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <a href="#CITEREFvon_Neumann1955">von Neumann 1955</a>, Theorem 16 </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> <a href="#CITEREFvon_Neumann1955">von Neumann 1955</a>, Theorem 14 </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> <a href="#CITEREFKakutani1939">Kakutani 1939</a> </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> <a href="#CITEREFLindenstraussTzafriri1971">Lindenstrauss & Tzafriri 1971</a> </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> <a href="#CITEREFHalmos1957">Halmos 1957</a>, §12 </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> A general account of spectral theory in Hilbert spaces can be found in <a href="#CITEREFRieszSz.-Nagy1990">Riesz & Sz.-Nagy (1990)</a>. A more sophisticated account in the language of C*-algebras is in <a href="#CITEREFRudin1973">Rudin (1973)</a> or <a href="#CITEREFKadisonRingrose1997">Kadison & Ringrose (1997)</a> </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> See, for instance, <a href="#CITEREFRieszSz.-Nagy1990">Riesz & Sz.-Nagy (1990</a>, Chapter VI) or <a href="#CITEREFWeidmann1980">Weidmann 1980</a>, Chapter 7. This result was already known to <a href="#CITEREFSchmidt1908">Schmidt (1908)</a> in the case of operators arising from integral kernels. </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> <a href="#CITEREFRieszSz.-Nagy1990">Riesz & Sz.-Nagy 1990</a>, §§107–108 </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> <a href="#CITEREFShubin1987">Shubin 1987</a> </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> <a href="#CITEREFRudin1973">Rudin 1973</a>, Theorem 13.30. </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPynchon1973" class="citation book cs1">Pynchon, Thomas (1973). Gravity's Rainbow. Viking Press. pp. 217, 275. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0143039945" title="Special:BookSources/978-0143039945"><bdi>978-0143039945</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravity%27s+Rainbow&rft.pages=217%2C+275&rft.pub=Viking+Press&rft.date=1973&rft.isbn=978-0143039945&rft.aulast=Pynchon&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=45" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler2014" class="citation cs2"><a href="/wiki/Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon</a> (18 December 2014), Linear Algebra Done Right, <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a> (3rd ed.), <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer Publishing</a> (published 2015), p. 296, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-11079-0" title="Special:BookSources/978-3-319-11079-0"><bdi>978-3-319-11079-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=296&rft.edition=3rd&rft.pub=Springer+Publishing&rft.date=2014-12-18&rft.isbn=978-3-319-11079-0&rft.aulast=Axler&rft.aufirst=Sheldon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachmanNariciBeckenstein2000" class="citation cs2">Bachman, George; Narici, Lawrence; Beckenstein, Edward (2000), Fourier and wavelet analysis, Universitext, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98899-3" title="Special:BookSources/978-0-387-98899-3"><bdi>978-0-387-98899-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1729490">1729490</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+and+wavelet+analysis&rft.place=Berlin%2C+New+York&rft.series=Universitext&rft.pub=Springer-Verlag&rft.date=2000&rft.isbn=978-0-387-98899-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1729490%23id-name%3DMR&rft.aulast=Bachman&rft.aufirst=George&rft.au=Narici%2C+Lawrence&rft.au=Beckenstein%2C+Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBersJohnSchechter1981" class="citation cs2"><a href="/wiki/Lipman_Bers" title="Lipman Bers">Bers, Lipman</a>; <a href="/wiki/Fritz_John" title="Fritz John">John, Fritz</a>; Schechter, Martin (1981), Partial differential equations, American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0049-2" title="Special:BookSources/978-0-8218-0049-2"><bdi>978-0-8218-0049-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+differential+equations&rft.pub=American+Mathematical+Society&rft.date=1981&rft.isbn=978-0-8218-0049-2&rft.aulast=Bers&rft.aufirst=Lipman&rft.au=John%2C+Fritz&rft.au=Schechter%2C+Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBillingsley1986" class="citation cs2">Billingsley, Patrick (1986), Probability and measure, Wiley</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+measure&rft.pub=Wiley&rft.date=1986&rft.aulast=Billingsley&rft.aufirst=Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1986" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1986), Spectral theories, Elements of mathematics, Berlin: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-00767-1" title="Special:BookSources/978-0-201-00767-1"><bdi>978-0-201-00767-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spectral+theories&rft.place=Berlin&rft.series=Elements+of+mathematics&rft.pub=Springer-Verlag&rft.date=1986&rft.isbn=978-0-201-00767-1&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1987" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-13627-9" title="Special:BookSources/978-3-540-13627-9"><bdi>978-3-540-13627-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+vector+spaces&rft.place=Berlin&rft.series=Elements+of+mathematics&rft.pub=Springer-Verlag&rft.date=1987&rft.isbn=978-3-540-13627-9&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyerMerzbach1991" class="citation cs2"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl Benjamin</a>; <a href="/wiki/Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C</a> (1991), <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye">A History of Mathematics</a> (2nd ed.), John Wiley & Sons, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=Carl+Benjamin&rft.au=Merzbach%2C+Uta+C&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrennerScott2005" class="citation cs2">Brenner, S.; Scott, R. L. (2005), The Mathematical Theory of Finite Element Methods (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95451-6" title="Special:BookSources/978-0-387-95451-6"><bdi>978-0-387-95451-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Theory+of+Finite+Element+Methods&rft.edition=2nd&rft.pub=Springer&rft.date=2005&rft.isbn=978-0-387-95451-6&rft.aulast=Brenner&rft.aufirst=S.&rft.au=Scott%2C+R.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrezis2010" class="citation cs2"><a href="/wiki/Haim_Brezis" class="mw-redirect" title="Haim Brezis">Brezis, Haim</a> (2010), Functional analysis, Sobolev spaces, and partial differential equations, Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+analysis%2C+Sobolev+spaces%2C+and+partial+differential+equations&rft.pub=Springer&rft.date=2010&rft.aulast=Brezis&rft.aufirst=Haim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFButtazzoGiaquintaHildebrandt1998" class="citation cs2">Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan (1998), One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications, vol. 15, The Clarendon Press Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850465-8" title="Special:BookSources/978-0-19-850465-8"><bdi>978-0-19-850465-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1694383">1694383</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=One-dimensional+variational+problems&rft.series=Oxford+Lecture+Series+in+Mathematics+and+its+Applications&rft.pub=The+Clarendon+Press+Oxford+University+Press&rft.date=1998&rft.isbn=978-0-19-850465-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1694383%23id-name%3DMR&rft.aulast=Buttazzo&rft.aufirst=Giuseppe&rft.au=Giaquinta%2C+Mariano&rft.au=Hildebrandt%2C+Stefan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClarkson1936" class="citation cs2">Clarkson, J. A. (1936), "Uniformly convex spaces", Trans. Amer. Math. Soc., 40 (3): 396–414, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1989630">10.2307/1989630</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1989630">1989630</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trans.+Amer.+Math.+Soc.&rft.atitle=Uniformly+convex+spaces&rft.volume=40&rft.issue=3&rft.pages=396-414&rft.date=1936&rft_id=info%3Adoi%2F10.2307%2F1989630&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1989630%23id-name%3DJSTOR&rft.aulast=Clarkson&rft.aufirst=J.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourantHilbert1953" class="citation cs2"><a href="/wiki/Richard_Courant" title="Richard Courant">Courant, Richard</a>; <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a> (1953), Methods of Mathematical Physics, Vol. I, Interscience</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Mathematical+Physics%2C+Vol.+I&rft.pub=Interscience&rft.date=1953&rft.aulast=Courant&rft.aufirst=Richard&rft.au=Hilbert%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieudonné1960" class="citation cs2"><a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1960), Foundations of Modern Analysis, Academic Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Modern+Analysis&rft.pub=Academic+Press&rft.date=1960&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac1930" class="citation cs2"><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac, P.A.M.</a> (1930), <a href="/wiki/The_Principles_of_Quantum_Mechanics" title="The Principles of Quantum Mechanics">The Principles of Quantum Mechanics</a>, Oxford: Clarendon Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Principles+of+Quantum+Mechanics&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=1930&rft.aulast=Dirac&rft.aufirst=P.A.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunfordSchwartz1958" class="citation cs2">Dunford, N.; <a href="/wiki/Jacob_T._Schwartz" title="Jacob T. Schwartz">Schwartz, J.T.</a> (1958), Linear operators, Parts I and II, Wiley-Interscience</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+operators%2C+Parts+I+and+II&rft.pub=Wiley-Interscience&rft.date=1958&rft.aulast=Dunford&rft.aufirst=N.&rft.au=Schwartz%2C+J.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuren1970" class="citation cs2">Duren, P. (1970), Theory of Hp-Spaces, New York: Academic Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+H%3Csup%3Ep%3C%2Fsup%3E-Spaces&rft.place=New+York&rft.pub=Academic+Press&rft.date=1970&rft.aulast=Duren&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinsiedlerWard2011" class="citation cs2">Einsiedler, Manfred; Ward, Thomas (2011), Ergodic theory with a view towards number theory, Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ergodic+theory+with+a+view+towards+number+theory&rft.pub=Springer&rft.date=2011&rft.aulast=Einsiedler&rft.aufirst=Manfred&rft.au=Ward%2C+Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans1998" class="citation cs2"><a href="/wiki/Lawrence_C._Evans" title="Lawrence C. Evans">Evans, L. C.</a> (1998), Partial Differential Equations, Providence: American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0772-2" title="Special:BookSources/0-8218-0772-2"><bdi>0-8218-0772-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations&rft.place=Providence&rft.pub=American+Mathematical+Society&rft.date=1998&rft.isbn=0-8218-0772-2&rft.aulast=Evans&rft.aufirst=L.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland2009" class="citation cs2">Folland, Gerald B. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?as_isbn=0-8218-4790-2">Fourier analysis and its application</a> (Reprint of Wadsworth and Brooks/Cole 1992 ed.), American Mathematical Society Bookstore, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4790-9" title="Special:BookSources/978-0-8218-4790-9"><bdi>978-0-8218-4790-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+analysis+and+its+application&rft.edition=Reprint+of+Wadsworth+and+Brooks%2FCole+1992&rft.pub=American+Mathematical+Society+Bookstore&rft.date=2009&rft.isbn=978-0-8218-4790-9&rft.aulast=Folland&rft.aufirst=Gerald+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fas_isbn%3D0-8218-4790-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland1989" class="citation cs2">Folland, Gerald B. (1989), Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08527-2" title="Special:BookSources/978-0-691-08527-2"><bdi>978-0-691-08527-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Harmonic+analysis+in+phase+space&rft.series=Annals+of+Mathematics+Studies&rft.pub=Princeton+University+Press&rft.date=1989&rft.isbn=978-0-691-08527-2&rft.aulast=Folland&rft.aufirst=Gerald+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFréchet1907" class="citation cs2">Fréchet, Maurice (1907), "Sur les ensembles de fonctions et les opérations linéaires", C. R. Acad. Sci. Paris, 144: 1414–1416</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=C.+R.+Acad.+Sci.+Paris&rft.atitle=Sur+les+ensembles+de+fonctions+et+les+op%C3%A9rations+lin%C3%A9aires&rft.volume=144&rft.pages=1414-1416&rft.date=1907&rft.aulast=Fr%C3%A9chet&rft.aufirst=Maurice&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFréchet1904" class="citation cs2">Fréchet, Maurice (1904), "Sur les opérations linéaires", Transactions of the American Mathematical Society, 5 (4): 493–499, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1986278">10.2307/1986278</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1986278">1986278</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Sur+les+op%C3%A9rations+lin%C3%A9aires&rft.volume=5&rft.issue=4&rft.pages=493-499&rft.date=1904&rft_id=info%3Adoi%2F10.2307%2F1986278&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1986278%23id-name%3DJSTOR&rft.aulast=Fr%C3%A9chet&rft.aufirst=Maurice&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGiusti2003" class="citation cs2">Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, World Scientific, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-043-2" title="Special:BookSources/978-981-238-043-2"><bdi>978-981-238-043-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Direct+Methods+in+the+Calculus+of+Variations&rft.pub=World+Scientific&rft.date=2003&rft.isbn=978-981-238-043-2&rft.aulast=Giusti&rft.aufirst=Enrico&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrattan-Guinness2000" class="citation cs2">Grattan-Guinness, Ivor (2000), The search for mathematical roots, 1870–1940, Princeton Paperbacks, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-05858-0" title="Special:BookSources/978-0-691-05858-0"><bdi>978-0-691-05858-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1807717">1807717</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+search+for+mathematical+roots%2C+1870%E2%80%931940&rft.series=Princeton+Paperbacks&rft.pub=Princeton+University+Press&rft.date=2000&rft.isbn=978-0-691-05858-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1807717%23id-name%3DMR&rft.aulast=Grattan-Guinness&rft.aufirst=Ivor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1957" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul</a> (1957), Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea Pub. Co</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Hilbert+Space+and+the+Theory+of+Spectral+Multiplicity&rft.pub=Chelsea+Pub.+Co&rft.date=1957&rft.aulast=Halmos&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1982" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul</a> (1982), A Hilbert Space Problem Book, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90685-0" title="Special:BookSources/978-0-387-90685-0"><bdi>978-0-387-90685-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Hilbert+Space+Problem+Book&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=978-0-387-90685-0&rft.aulast=Halmos&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHewittStromberg1965" class="citation cs2">Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis, New York: Springer-Verlag</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Abstract+Analysis&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1965&rft.aulast=Hewitt&rft.aufirst=Edwin&rft.au=Stromberg%2C+Karl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbertNordheimvon_Neumann1927" class="citation cs2"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a>; <a href="/wiki/Lothar_Nordheim" class="mw-redirect" title="Lothar Nordheim">Nordheim, Lothar Wolfgang</a>; <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (1927), "Über die Grundlagen der Quantenmechanik", Mathematische Annalen, 98: 1–30, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01451579">10.1007/BF01451579</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120986758">120986758</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=%C3%9Cber+die+Grundlagen+der+Quantenmechanik&rft.volume=98&rft.pages=1-30&rft.date=1927&rft_id=info%3Adoi%2F10.1007%2FBF01451579&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120986758%23id-name%3DS2CID&rft.aulast=Hilbert&rft.aufirst=David&rft.au=Nordheim%2C+Lothar+Wolfgang&rft.au=von+Neumann%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolevo2001" class="citation cs2"><a href="/wiki/Alexander_Holevo" title="Alexander Holevo">Holevo, Alexander S.</a> (2001), Statistical Structure of Quantum Theory, Lecture Notes in Physics, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-42082-7" title="Special:BookSources/3-540-42082-7"><bdi>3-540-42082-7</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/318268606">318268606</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Structure+of+Quantum+Theory&rft.series=Lecture+Notes+in+Physics&rft.pub=Springer&rft.date=2001&rft_id=info%3Aoclcnum%2F318268606&rft.isbn=3-540-42082-7&rft.aulast=Holevo&rft.aufirst=Alexander+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKac1966" class="citation cs2"><a href="/wiki/Mark_Kac" title="Mark Kac">Kac, Mark</a> (1966), "Can one hear the shape of a drum?", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>, 73 (4, part 2): 1–23, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2313748">10.2307/2313748</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2313748">2313748</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Can+one+hear+the+shape+of+a+drum%3F&rft.volume=73&rft.issue=4%2C+part+2&rft.pages=1-23&rft.date=1966&rft_id=info%3Adoi%2F10.2307%2F2313748&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2313748%23id-name%3DJSTOR&rft.aulast=Kac&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKadisonRingrose1997" class="citation cs2">Kadison, Richard V.; Ringrose, John R. (1997), Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0819-1" title="Special:BookSources/978-0-8218-0819-1"><bdi>978-0-8218-0819-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1468229">1468229</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+the+theory+of+operator+algebras.+Vol.+I&rft.place=Providence%2C+R.I.&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=1997&rft.isbn=978-0-8218-0819-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1468229%23id-name%3DMR&rft.aulast=Kadison&rft.aufirst=Richard+V.&rft.au=Ringrose%2C+John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKadisonRingrose1983" class="citation cs2">Kadison, Richard V.; Ringrose, John R. (1983), Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory, New York: Academic Press, Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+the+Theory+of+Operator+Algebras%2C+Vol.+I%3A+Elementary+Theory&rft.place=New+York&rft.pub=Academic+Press%2C+Inc.&rft.date=1983&rft.aulast=Kadison&rft.aufirst=Richard+V.&rft.au=Ringrose%2C+John+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaratzasShreve2019" class="citation cs2">Karatzas, Ioannis; Shreve, Steven (2019), Brownian Motion and Stochastic Calculus (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97655-6" title="Special:BookSources/978-0-387-97655-6"><bdi>978-0-387-97655-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Brownian+Motion+and+Stochastic+Calculus&rft.edition=2nd&rft.pub=Springer&rft.date=2019&rft.isbn=978-0-387-97655-6&rft.aulast=Karatzas&rft.aufirst=Ioannis&rft.au=Shreve%2C+Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKakutani1939" class="citation cs2"><a href="/wiki/Shizuo_Kakutani" title="Shizuo Kakutani">Kakutani, Shizuo</a> (1939), "Some characterizations of Euclidean space", Japanese Journal of Mathematics, 16: 93–97, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4099%2Fjjm1924.16.0_93">10.4099/jjm1924.16.0_93</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0000895">0000895</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Japanese+Journal+of+Mathematics&rft.atitle=Some+characterizations+of+Euclidean+space&rft.volume=16&rft.pages=93-97&rft.date=1939&rft_id=info%3Adoi%2F10.4099%2Fjjm1924.16.0_93&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0000895%23id-name%3DMR&rft.aulast=Kakutani&rft.aufirst=Shizuo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1972" class="citation cs2"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a> (1972), <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalthou00morr">Mathematical thought from ancient to modern times, Volume 3</a> (3rd ed.), <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a> (published 1990), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-506137-6" title="Special:BookSources/978-0-19-506137-6"><bdi>978-0-19-506137-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+thought+from+ancient+to+modern+times%2C+Volume+3&rft.edition=3rd&rft.pub=Oxford+University+Press&rft.date=1972&rft.isbn=978-0-19-506137-6&rft.aulast=Kline&rft.aufirst=Morris&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalthou00morr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolmogorovFomin1970" class="citation cs2"><a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov, Andrey</a>; <a href="/wiki/Sergei_Fomin" title="Sergei Fomin">Fomin, Sergei V.</a> (1970), <a rel="nofollow" class="external text" href="https://archive.org/details/introductoryreal00kolm_0">Introductory Real Analysis</a> (Revised English edition, trans. by Richard A. Silverman (1975) ed.), Dover Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61226-3" title="Special:BookSources/978-0-486-61226-3"><bdi>978-0-486-61226-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Real+Analysis&rft.edition=Revised+English+edition%2C+trans.+by+Richard+A.+Silverman+%281975%29&rft.pub=Dover+Press&rft.date=1970&rft.isbn=978-0-486-61226-3&rft.aulast=Kolmogorov&rft.aufirst=Andrey&rft.au=Fomin%2C+Sergei+V.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductoryreal00kolm_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrantz2002" class="citation cs2"><a href="/wiki/Steven_Krantz" class="mw-redirect" title="Steven Krantz">Krantz, Steven G.</a> (2002), Function Theory of Several Complex Variables, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2724-6" title="Special:BookSources/978-0-8218-2724-6"><bdi>978-0-8218-2724-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Function+Theory+of+Several+Complex+Variables&rft.place=Providence%2C+R.I.&rft.pub=American+Mathematical+Society&rft.date=2002&rft.isbn=978-0-8218-2724-6&rft.aulast=Krantz&rft.aufirst=Steven+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanczos1988" class="citation cs2">Lanczos, Cornelius (1988), <a rel="nofollow" class="external text" href="https://books.google.com/books?as_isbn=0-486-65656-X">Applied analysis</a> (Reprint of 1956 Prentice-Hall ed.), Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65656-4" title="Special:BookSources/978-0-486-65656-4"><bdi>978-0-486-65656-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+analysis&rft.edition=Reprint+of+1956+Prentice-Hall&rft.pub=Dover+Publications&rft.date=1988&rft.isbn=978-0-486-65656-4&rft.aulast=Lanczos&rft.aufirst=Cornelius&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fas_isbn%3D0-486-65656-X&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLebesgue1904" class="citation cs2">Lebesgue, Henri (1904), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VfUKAAAAYAAJ&q=%22Lebesgue%22%20%22Le%C3%A7ons%20sur%20l'int%C3%A9gration%20et%20la%20recherche%20des%20fonctions%20...%22&pg=PA1">Leçons sur l'intégration et la recherche des fonctions primitives</a>, Gauthier-Villars</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le%C3%A7ons+sur+l%27int%C3%A9gration+et+la+recherche+des+fonctions+primitives&rft.pub=Gauthier-Villars&rft.date=1904&rft.aulast=Lebesgue&rft.aufirst=Henri&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVfUKAAAAYAAJ%26q%3D%2522Lebesgue%2522%2520%2522Le%25C3%25A7ons%2520sur%2520l%27int%25C3%25A9gration%2520et%2520la%2520recherche%2520des%2520fonctions%2520...%2522%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevitan2001" class="citation cs2">Levitan, B.M. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Hilbert_space">"Hilbert space"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hilbert+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Levitan&rft.aufirst=B.M.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DHilbert_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLindenstraussTzafriri1971" class="citation cs2">Lindenstrauss, J.; Tzafriri, L. (1971), "On the complemented subspaces problem", <a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a>, 9 (2): 263–269, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02771592">10.1007/BF02771592</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-2172">0021-2172</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0276734">0276734</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119575718">119575718</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Israel+Journal+of+Mathematics&rft.atitle=On+the+complemented+subspaces+problem&rft.volume=9&rft.issue=2&rft.pages=263-269&rft.date=1971&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119575718%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0276734%23id-name%3DMR&rft.issn=0021-2172&rft_id=info%3Adoi%2F10.1007%2FBF02771592&rft.aulast=Lindenstrauss&rft.aufirst=J.&rft.au=Tzafriri%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarsden1974" class="citation cs2"><a href="/wiki/Jerrold_E._Marsden" title="Jerrold E. Marsden">Marsden, Jerrold E.</a> (1974), Elementary classical analysis, W. H. Freeman and Co., <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0357693">0357693</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+classical+analysis&rft.pub=W.+H.+Freeman+and+Co.&rft.date=1974&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0357693%23id-name%3DMR&rft.aulast=Marsden&rft.aufirst=Jerrold+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurphy1990" class="citation cs2">Murphy, Gerald J. (1990), C*-algebras and Operator Theory, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-511360-9" title="Special:BookSources/0-12-511360-9"><bdi>0-12-511360-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C%2A-algebras+and+Operator+Theory&rft.pub=Academic+Press&rft.date=1990&rft.isbn=0-12-511360-9&rft.aulast=Murphy&rft.aufirst=Gerald+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1929" class="citation cs2"><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (1929), "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren", Mathematische Annalen, 102: 49–131, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01782338">10.1007/BF01782338</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121249803">121249803</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Allgemeine+Eigenwerttheorie+Hermitescher+Funktionaloperatoren&rft.volume=102&rft.pages=49-131&rft.date=1929&rft_id=info%3Adoi%2F10.1007%2FBF01782338&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121249803%23id-name%3DS2CID&rft.aulast=von+Neumann&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1932" class="citation cs2"><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA, 18 (3): 263–266, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1932PNAS...18..263N">1932PNAS...18..263N</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.18.3.263">10.1073/pnas.18.3.263</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/86260">86260</a>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076204">1076204</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16587674">16587674</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc+Natl+Acad+Sci+USA&rft.atitle=Physical+Applications+of+the+Ergodic+Hypothesis&rft.volume=18&rft.issue=3&rft.pages=263-266&rft.date=1932&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1076204%23id-name%3DPMC&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F86260%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1932PNAS...18..263N&rft_id=info%3Apmid%2F16587674&rft_id=info%3Adoi%2F10.1073%2Fpnas.18.3.263&rft.aulast=von+Neumann&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1955" class="citation cs2"><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann, John</a> (1955), <a href="/wiki/Mathematical_Foundations_of_Quantum_Mechanics" title="Mathematical Foundations of Quantum Mechanics">Mathematical Foundations of Quantum Mechanics</a>, Princeton Landmarks in Mathematics, translated by Beyer, Robert T., <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a> (published 1996), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02893-4" title="Special:BookSources/978-0-691-02893-4"><bdi>978-0-691-02893-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1435976">1435976</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Foundations+of+Quantum+Mechanics&rft.series=Princeton+Landmarks+in+Mathematics&rft.pub=Princeton+University+Press&rft.date=1955&rft.isbn=978-0-691-02893-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1435976%23id-name%3DMR&rft.aulast=von+Neumann&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsenChuang2000" class="citation cs2"><a href="/wiki/Michael_Nielsen" title="Michael Nielsen">Nielsen, Michael A.</a>; <a href="/wiki/Isaac_Chuang" title="Isaac Chuang">Chuang, Isaac L.</a> (2000), <a href="/wiki/Quantum_Computation_and_Quantum_Information" title="Quantum Computation and Quantum Information">Quantum Computation and Quantum Information</a> (1st ed.), Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-63503-5" title="Special:BookSources/978-0-521-63503-5"><bdi>978-0-521-63503-5</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/634735192">634735192</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computation+and+Quantum+Information&rft.place=Cambridge&rft.edition=1st&rft.pub=Cambridge+University+Press&rft.date=2000&rft_id=info%3Aoclcnum%2F634735192&rft.isbn=978-0-521-63503-5&rft.aulast=Nielsen&rft.aufirst=Michael+A.&rft.au=Chuang%2C+Isaac+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson1996" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a> (1996), <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Abstract_linear_spaces.html">"Abstract linear spaces"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abstract+linear+spaces&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.date=1996&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FAbstract_linear_spaces.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPathria1996" class="citation cs2">Pathria, RK (1996), Statistical mechanics (2 ed.), Academic Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+mechanics&rft.edition=2&rft.pub=Academic+Press&rft.date=1996&rft.aulast=Pathria&rft.aufirst=RK&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPedersen1995" class="citation cs2">Pedersen, Gert (1995), Analysis Now, Graduate Texts in Mathematics, vol. 118, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-6981-6" title="Special:BookSources/978-1-4612-6981-6"><bdi>978-1-4612-6981-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0971256">0971256</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+Now&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-1-4612-6981-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0971256%23id-name%3DMR&rft.aulast=Pedersen&rft.aufirst=Gert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeres1993" class="citation cs2"><a href="/wiki/Asher_Peres" title="Asher Peres">Peres, Asher</a> (1993), <a href="/wiki/Quantum_Theory:_Concepts_and_Methods" title="Quantum Theory: Concepts and Methods">Quantum Theory: Concepts and Methods</a>, Kluwer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-2549-4" title="Special:BookSources/0-7923-2549-4"><bdi>0-7923-2549-4</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/28854083">28854083</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory%3A+Concepts+and+Methods&rft.pub=Kluwer&rft.date=1993&rft_id=info%3Aoclcnum%2F28854083&rft.isbn=0-7923-2549-4&rft.aulast=Peres&rft.aufirst=Asher&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrugovečki1981" class="citation cs2">Prugovečki, Eduard (1981), Quantum mechanics in Hilbert space (2nd ed.), Dover (published 2006), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45327-9" title="Special:BookSources/978-0-486-45327-9"><bdi>978-0-486-45327-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+mechanics+in+Hilbert+space&rft.edition=2nd&rft.pub=Dover&rft.date=1981&rft.isbn=978-0-486-45327-9&rft.aulast=Prugove%C4%8Dki&rft.aufirst=Eduard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReedSimon1980" class="citation cs2"><a href="/wiki/Michael_C._Reed" title="Michael C. Reed">Reed, Michael</a>; <a href="/wiki/Barry_Simon" title="Barry Simon">Simon, Barry</a> (1980), Functional Analysis (vol I of 4 vols), Methods of Modern Mathematical Physics, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-585050-6" title="Special:BookSources/978-0-12-585050-6"><bdi>978-0-12-585050-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis+%28vol+I+of+4+vols%29&rft.series=Methods+of+Modern+Mathematical+Physics&rft.pub=Academic+Press&rft.date=1980&rft.isbn=978-0-12-585050-6&rft.aulast=Reed&rft.aufirst=Michael&rft.au=Simon%2C+Barry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReedSimon1975" class="citation cs2"><a href="/wiki/Michael_C._Reed" title="Michael C. Reed">Reed, Michael</a>; <a href="/wiki/Barry_Simon" title="Barry Simon">Simon, Barry</a> (1975), Fourier Analysis, Self-Adjointness (vol II of 4 vols), Methods of Modern Mathematical Physics, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780125850025" title="Special:BookSources/9780125850025"><bdi>9780125850025</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Analysis%2C+Self-Adjointness+%28vol+II+of+4+vols%29&rft.series=Methods+of+Modern+Mathematical+Physics&rft.pub=Academic+Press&rft.date=1975&rft.isbn=9780125850025&rft.aulast=Reed&rft.aufirst=Michael&rft.au=Simon%2C+Barry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRieffelPolak2011" class="citation cs2"><a href="/wiki/Eleanor_Rieffel" title="Eleanor Rieffel">Rieffel, Eleanor G.</a>; Polak, Wolfgang H. (2011-03-04), <a href="/wiki/Quantum_Computing:_A_Gentle_Introduction" title="Quantum Computing: A Gentle Introduction">Quantum Computing: A Gentle Introduction</a>, MIT Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-01506-6" title="Special:BookSources/978-0-262-01506-6"><bdi>978-0-262-01506-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing%3A+A+Gentle+Introduction&rft.pub=MIT+Press&rft.date=2011-03-04&rft.isbn=978-0-262-01506-6&rft.aulast=Rieffel&rft.aufirst=Eleanor+G.&rft.au=Polak%2C+Wolfgang+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiesz1907" class="citation cs2"><a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Riesz, Frigyes</a> (1907), "Sur une espèce de Géométrie analytique des systèmes de fonctions sommables", C. R. Acad. Sci. Paris, 144: 1409–1411</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=C.+R.+Acad.+Sci.+Paris&rft.atitle=Sur+une+esp%C3%A8ce+de+G%C3%A9om%C3%A9trie+analytique+des+syst%C3%A8mes+de+fonctions+sommables&rft.volume=144&rft.pages=1409-1411&rft.date=1907&rft.aulast=Riesz&rft.aufirst=Frigyes&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiesz1934" class="citation cs2"><a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Riesz, Frigyes</a> (1934), "Zur Theorie des Hilbertschen Raumes", Acta Sci. Math. Szeged, 7: 34–38</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Sci.+Math.+Szeged&rft.atitle=Zur+Theorie+des+Hilbertschen+Raumes&rft.volume=7&rft.pages=34-38&rft.date=1934&rft.aulast=Riesz&rft.aufirst=Frigyes&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRieszSz.-Nagy1990" class="citation cs2"><a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Riesz, Frigyes</a>; <a href="/wiki/B%C3%A9la_Sz%C5%91kefalvi-Nagy" title="Béla Szőkefalvi-Nagy">Sz.-Nagy, Béla</a> (1990), Functional analysis, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66289-3" title="Special:BookSources/978-0-486-66289-3"><bdi>978-0-486-66289-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+analysis&rft.pub=Dover&rft.date=1990&rft.isbn=978-0-486-66289-3&rft.aulast=Riesz&rft.aufirst=Frigyes&rft.au=Sz.-Nagy%2C+B%C3%A9la&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoman2008" class="citation cs2"><a href="/wiki/Steven_Roman" title="Steven Roman">Roman, Stephen</a> (2008), Advanced Linear Algebra, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a> (Third ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-72828-5" title="Special:BookSources/978-0-387-72828-5"><bdi>978-0-387-72828-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Linear+Algebra&rft.series=Graduate+Texts+in+Mathematics&rft.edition=Third&rft.pub=Springer&rft.date=2008&rft.isbn=978-0-387-72828-5&rft.aulast=Roman&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1973" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1973). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: <a href="/wiki/McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780070542259" title="Special:BookSources/9780070542259"><bdi>9780070542259</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.place=New+York%2C+NY&rft.series=International+Series+in+Pure+and+Applied+Mathematics&rft.edition=First&rft.pub=McGraw-Hill+Science%2FEngineering%2FMath&rft.date=1973&rft.isbn=9780070542259&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1987" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1987), Real and Complex Analysis, McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-100276-9" title="Special:BookSources/978-0-07-100276-9"><bdi>978-0-07-100276-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Complex+Analysis&rft.pub=McGraw-Hill&rft.date=1987&rft.isbn=978-0-07-100276-9&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaks2005" class="citation cs2"><a href="/wiki/Stanis%C5%82aw_Saks" title="Stanisław Saks">Saks, Stanisław</a> (2005), Theory of the integral (2nd Dover ed.), Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-44648-6" title="Special:BookSources/978-0-486-44648-6"><bdi>978-0-486-44648-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+the+integral&rft.edition=2nd+Dover&rft.pub=Dover&rft.date=2005&rft.isbn=978-0-486-44648-6&rft.aulast=Saks&rft.aufirst=Stanis%C5%82aw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">; originally published Monografje Matematyczne, vol. 7, Warszawa, 1937.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="/wiki/Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-7155-0" title="Special:BookSources/978-1-4612-7155-0"><bdi>978-1-4612-7155-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=New+York%2C+NY&rft.series=GTM&rft.edition=Second&rft.pub=Springer+New+York+Imprint+Springer&rft.date=1999&rft_id=info%3Aoclcnum%2F840278135&rft.isbn=978-1-4612-7155-0&rft.aulast=Schaefer&rft.aufirst=Helmut+H.&rft.au=Wolff%2C+Manfred+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt1908" class="citation cs2"><a href="/wiki/Erhard_Schmidt" title="Erhard Schmidt">Schmidt, Erhard</a> (1908), "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten", Rend. Circ. Mat. Palermo, 25: 63–77, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03029116">10.1007/BF03029116</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120666844">120666844</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rend.+Circ.+Mat.+Palermo&rft.atitle=%C3%9Cber+die+Aufl%C3%B6sung+linearer+Gleichungen+mit+unendlich+vielen+Unbekannten&rft.volume=25&rft.pages=63-77&rft.date=1908&rft_id=info%3Adoi%2F10.1007%2FBF03029116&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120666844%23id-name%3DS2CID&rft.aulast=Schmidt&rft.aufirst=Erhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShubin1987" class="citation cs2">Shubin, M. A. (1987), Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-13621-7" title="Special:BookSources/978-3-540-13621-7"><bdi>978-3-540-13621-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0883081">0883081</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pseudodifferential+operators+and+spectral+theory&rft.place=Berlin%2C+New+York&rft.series=Springer+Series+in+Soviet+Mathematics&rft.pub=Springer-Verlag&rft.date=1987&rft.isbn=978-3-540-13621-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D883081%23id-name%3DMR&rft.aulast=Shubin&rft.aufirst=M.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSobrino1996" class="citation cs2">Sobrino, Luis (1996), Elements of non-relativistic quantum mechanics, River Edge, New Jersey: World Scientific Publishing Co. Inc., <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996lnrq.book.....S">1996lnrq.book.....S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F2865">10.1142/2865</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-2386-1" title="Special:BookSources/978-981-02-2386-1"><bdi>978-981-02-2386-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1626401">1626401</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+non-relativistic+quantum+mechanics&rft.place=River+Edge%2C+New+Jersey&rft.pub=World+Scientific+Publishing+Co.+Inc.&rft.date=1996&rft_id=info%3Adoi%2F10.1142%2F2865&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1626401%23id-name%3DMR&rft_id=info%3Abibcode%2F1996lnrq.book.....S&rft.isbn=978-981-02-2386-1&rft.aulast=Sobrino&rft.aufirst=Luis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStapleton1995" class="citation cs2">Stapleton, James (1995), Linear statistical models, John Wiley and Sons</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+statistical+models&rft.pub=John+Wiley+and+Sons&rft.date=1995&rft.aulast=Stapleton&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart2006" class="citation cs2">Stewart, James (2006), Calculus: Concepts and Contexts (3rd ed.), Thomson/Brooks/Cole</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Concepts+and+Contexts&rft.edition=3rd&rft.pub=Thomson%2FBrooks%2FCole&rft.date=2006&rft.aulast=Stewart&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStein1970" class="citation cs2">Stein, E (1970), <a rel="nofollow" class="external text" href="https://archive.org/details/singularintegral0000stei">Singular Integrals and Differentiability Properties of Functions</a>, Princeton Univ. Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08079-6" title="Special:BookSources/978-0-691-08079-6"><bdi>978-0-691-08079-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Singular+Integrals+and+Differentiability+Properties+of+Functions&rft.pub=Princeton+Univ.+Press&rft.date=1970&rft.isbn=978-0-691-08079-6&rft.aulast=Stein&rft.aufirst=E&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsingularintegral0000stei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinWeiss1971" class="citation cs2"><a href="/wiki/Elias_Stein" class="mw-redirect" title="Elias Stein">Stein, Elias</a>; <a href="/wiki/Guido_Weiss" title="Guido Weiss">Weiss, Guido</a> (1971), <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontofo0000stei">Introduction to Fourier Analysis on Euclidean Spaces</a>, Princeton, N.J.: Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08078-9" title="Special:BookSources/978-0-691-08078-9"><bdi>978-0-691-08078-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Fourier+Analysis+on+Euclidean+Spaces&rft.place=Princeton%2C+N.J.&rft.pub=Princeton+University+Press&rft.date=1971&rft.isbn=978-0-691-08078-9&rft.aulast=Stein&rft.aufirst=Elias&rft.au=Weiss%2C+Guido&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontofo0000stei&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinShakarchi2005" class="citation cs2">Stein, E; Shakarchi, R (2005), Real analysis, measure theory, integration, and Hilbert spaces, Princeton University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+analysis%2C+measure+theory%2C+integration%2C+and+Hilbert+spaces&rft.pub=Princeton+University+Press&rft.date=2005&rft.aulast=Stein&rft.aufirst=E&rft.au=Shakarchi%2C+R&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStreaterWightman1964" class="citation cs2"><a href="/wiki/Ray_Streater" title="Ray Streater">Streater, Ray</a>; <a href="/wiki/Arthur_Wightman" title="Arthur Wightman">Wightman, Arthur</a> (1964), PCT, Spin and Statistics and All That, W. A. Benjamin, Inc</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=PCT%2C+Spin+and+Statistics+and+All+That&rft.pub=W.+A.+Benjamin%2C+Inc&rft.date=1964&rft.aulast=Streater&rft.aufirst=Ray&rft.au=Wightman%2C+Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStroock2011" class="citation cs2">Stroock, Daniel (2011), Probability theory: an analytic view (2 ed.), Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+theory%3A+an+analytic+view&rft.edition=2&rft.pub=Cambridge+University+Press&rft.date=2011&rft.aulast=Stroock&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTeschl2009" class="citation book cs1"><a href="/wiki/Gerald_Teschl" title="Gerald Teschl">Teschl, Gerald</a> (2009). <a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/">Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators</a>. <a href="/wiki/Providence,_Rhode_Island" title="Providence, Rhode Island">Providence</a>: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4660-5" title="Special:BookSources/978-0-8218-4660-5"><bdi>978-0-8218-4660-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+in+Quantum+Mechanics%3B+With+Applications+to+Schr%C3%B6dinger+Operators&rft.place=Providence&rft.pub=American+Mathematical+Society&rft.date=2009&rft.isbn=978-0-8218-4660-5&rft.aulast=Teschl&rft.aufirst=Gerald&rft_id=https%3A%2F%2Fwww.mat.univie.ac.at%2F~gerald%2Fftp%2Fbook-schroe%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTitchmarsh1946" class="citation cs2"><a href="/wiki/Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Titchmarsh, Edward Charles</a> (1946), Eigenfunction expansions, part 1, Oxford University: Clarendon Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Eigenfunction+expansions%2C+part+1&rft.place=Oxford+University&rft.pub=Clarendon+Press&rft.date=1946&rft.aulast=Titchmarsh&rft.aufirst=Edward+Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrèves1967" class="citation cs2">Trèves, François (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces%2C+Distributions+and+Kernels&rft.pub=Academic+Press&rft.date=1967&rft.aulast=Tr%C3%A8ves&rft.aufirst=Fran%C3%A7ois&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarner1983" class="citation cs2">Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90894-6" title="Special:BookSources/978-0-387-90894-6"><bdi>978-0-387-90894-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Differentiable+Manifolds+and+Lie+Groups&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1983&rft.isbn=978-0-387-90894-6&rft.aulast=Warner&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeidmann1980" class="citation cs2">Weidmann, Joachim (1980), Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90427-6" title="Special:BookSources/978-0-387-90427-6"><bdi>978-0-387-90427-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0566954">0566954</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+operators+in+Hilbert+spaces&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1980&rft.isbn=978-0-387-90427-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D566954%23id-name%3DMR&rft.aulast=Weidmann&rft.aufirst=Joachim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeyl1931" class="citation cs2"><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl, Hermann</a> (1931), The Theory of Groups and Quantum Mechanics (English 1950 ed.), Dover Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60269-1" title="Special:BookSources/978-0-486-60269-1"><bdi>978-0-486-60269-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Groups+and+Quantum+Mechanics&rft.edition=English+1950&rft.pub=Dover+Press&rft.date=1931&rft.isbn=978-0-486-60269-1&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoung1988" class="citation cs2">Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-33071-8" title="Special:BookSources/978-0-521-33071-8"><bdi>978-0-521-33071-8</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0645.46024">0645.46024</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+Hilbert+space&rft.pub=Cambridge+University+Press&rft.date=1988&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0645.46024%23id-name%3DZbl&rft.isbn=978-0-521-33071-8&rft.aulast=Young&rft.aufirst=Nicholas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988">.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Hilbert_space&action=edit&section=46" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Functional_Analysis/Hilbert_spaces" class="extiw" title="wikibooks:Functional Analysis/Hilbert spaces">Functional Analysis/Hilbert spaces</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Hilbert_space" class="extiw" title="commons:Category:Hilbert space">Hilbert space</a>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Hilbert_space">"Hilbert space"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hilbert+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DHilbert_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHilbert+space" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/HilbertSpace.html">Hilbert space at Mathworld</a></li> <li><a rel="nofollow" class="external text" href="http://terrytao.wordpress.com/2009/01/17/254a-notes-5-hilbert-spaces/">245B, notes 5: Hilbert spaces</a> by <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .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{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}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Hilbert_spaces" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Hilbert_space" title="Template:Hilbert space"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Hilbert_space" title="Template talk:Hilbert space"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Hilbert_space" title="Special:EditPage/Template:Hilbert space"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Hilbert_spaces" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Hilbert spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product</a> and <a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li> <li><a class="mw-selflink selflink">Hilbert space</a> and <a href="/wiki/Prehilbert_space" class="mw-redirect" title="Prehilbert space">Prehilbert space</a></li> <li><a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a></li> <li><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's inequality</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_projection_theorem" title="Hilbert projection theorem">Hilbert projection theorem</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a> (<a href="/wiki/Parallelogram_law#The_parallelogram_law_in_inner_product_spaces" title="Parallelogram law">Parallelogram law</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">Compact operator on Hilbert space</a></li> <li><a href="/wiki/Densely_defined_operator" title="Densely defined operator">Densely defined</a></li> <li><a href="/wiki/Sesquilinear_form#Hermitian_form" title="Sesquilinear form">Hermitian form</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">Sesquilinear form</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Cn(K) with K compact & n<∞</a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann F</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Lp_spaces" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Lp_spaces" title="Template:Lp spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Lp_spaces" title="Template talk:Lp spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Lp_spaces" title="Special:EditPage/Template:Lp spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Lp_spaces" style="font-size:114%;margin:0 4em"><a href="/wiki/Lp_space" title="Lp space">Lp spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a> & <a class="mw-selflink selflink">Hilbert spaces</a></li> <li><a href="/wiki/Lp_space" title="Lp space">Lp spaces</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a> <ul><li><a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue</a></li></ul></li> <li><a href="/wiki/Measure_space" title="Measure space">Measure space</a></li> <li><a href="/wiki/Measurable_space" title="Measurable space">Measurable space</a>/<a href="/wiki/Measurable_function" title="Measurable function">function</a></li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence spaces</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L1_space" class="mw-redirect" title="L1 space">L1 spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integrable_function" class="mw-redirect" title="Integrable function">Integrable function</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L2_space" class="mw-redirect" title="L2 space">L2 spaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a class="mw-selflink selflink">Hilbert space</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Square-integrable_function" title="Square-integrable function">Square-integrable function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L-infinity" title="L-infinity"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bounded_function" title="Bounded function">Bounded function</a></li> <li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a> <ul><li><a href="/wiki/Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">Essential</a></li></ul></li> <li><a href="/wiki/Uniform_norm" title="Uniform norm">Uniform norm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li> <li><a href="/wiki/Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">Convergence almost everywhere</a></li> <li><a href="/wiki/Convergence_in_measure" title="Convergence in measure">Convergence in measure</a></li> <li><a href="/wiki/Function_space" title="Function space">Function space</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Locally_integrable_function" title="Locally integrable function">Locally integrable function</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable function</a></li> <li><a href="/wiki/Symmetric_decreasing_rearrangement" title="Symmetric decreasing rearrangement">Symmetric decreasing rearrangement</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inequalities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babenko%E2%80%93Beckner_inequality" title="Babenko–Beckner inequality">Babenko–Beckner</a></li> <li><a href="/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's</a></li> <li><a href="/wiki/Clarkson%27s_inequalities" title="Clarkson's inequalities">Clarkson's</a></li> <li><a href="/wiki/Hanner%27s_inequalities" title="Hanner's inequalities">Hanner's</a></li> <li><a href="/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young</a></li> <li><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's</a></li> <li><a href="/wiki/Markov%27s_inequality" title="Markov's inequality">Markov's</a></li> <li><a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski</a></li> <li><a href="/wiki/Young%27s_convolution_inequality" title="Young's convolution inequality">Young's convolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_analysis" title="Category:Theorems in analysis">Results</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Marcinkiewicz_interpolation_theorem" title="Marcinkiewicz interpolation theorem">Marcinkiewicz interpolation theorem</a></li> <li><a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue</a></li> <li><a href="/wiki/Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a></li> <li><a href="/wiki/Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">For <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li> <li><a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> <ul><li><a href="/wiki/Milman%27s_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li> <li><a href="/wiki/Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li> <li><a href="/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li> <li><a href="/wiki/Vitale%27s_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Lorentz_space" title="Lorentz space">Lorentz space</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinorm</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev space</a></li> <li><a href="/wiki/*-algebra" title="*-algebra">*-algebra</a> <ul><li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann</a></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Functional_analysis_(topics_–_glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a class="mw-selflink selflink">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a>/<a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_operator" title="Integral operator">Integral operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Banach_space_topics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Banach_spaces" title="Template:Banach spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Banach_spaces" title="Template talk:Banach spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Banach_spaces" title="Special:EditPage/Template:Banach spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Banach_space_topics" style="font-size:114%;margin:0 4em"><a href="/wiki/Banach_space" title="Banach space">Banach space</a> topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of Banach spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a> <ul><li><a href="/wiki/List_of_Banach_spaces" title="List of Banach spaces">list</a></li></ul></li> <li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach lattice</a></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck </a></li> <li><a class="mw-selflink selflink">Hilbert</a> <ul><li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li></ul></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a>) <a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz</a></li> <li><a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li> <li>(<a href="/wiki/B-convex_space" title="B-convex space">B</a></li> <li><a href="/wiki/Strictly_convex_space" title="Strictly convex space">Strictly</a></li> <li><a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li> <li><a href="/wiki/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li>(<a href="/wiki/Injective_tensor_product" title="Injective tensor product">Injective</a></li> <li><a href="/wiki/Projective_tensor_product" title="Projective tensor product">Projective</a>) <a href="/wiki/Topological_tensor_product" title="Topological tensor product">Tensor product</a> (<a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">of Hilbert spaces</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Banach spaces are:</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/F-space" title="F-space">F-space</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a> <ul><li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame</a></li></ul></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a> <ul><li><a href="/wiki/Locally_convex_topological_vector_space#Definition_via_seminorms" title="Locally convex topological vector space">Seminorms</a>/<a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski functionals</a></li></ul></li> <li><a href="/wiki/Mackey_space" title="Mackey space">Mackey</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable</a></li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed</a> <ul><li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a></li></ul></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Function space Topologies</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach%E2%80%93Mazur_compactum" title="Banach–Mazur compactum">Banach–Mazur compactum</a></li> <li><a href="/wiki/Dual_topology" title="Dual topology">Dual</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual space</a> <ul><li><a href="/wiki/Dual_norm" title="Dual norm">Dual norm</a></li></ul></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator</a></li> <li><a href="/wiki/Ultraweak_topology" title="Ultraweak topology">Ultraweak</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak</a> <ul><li><a href="/wiki/Weak_topology_(polar_topology)" class="mw-redirect" title="Weak topology (polar topology)">polar</a></li> <li><a href="/wiki/Weak_operator_topology" title="Weak operator topology">operator</a></li></ul></li> <li><a href="/wiki/Strong_topology" title="Strong topology">Strong</a> <ul><li><a href="/wiki/Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">polar</a></li> <li><a href="/wiki/Strong_operator_topology" title="Strong operator topology">operator</a></li></ul></li> <li><a href="/wiki/Ultrastrong_topology" title="Ultrastrong topology">Ultrastrong</a></li> <li><a href="/wiki/Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Uniform convergence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">Linear operators</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">operator</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear</a></li></ul></li> <li>(<a href="/wiki/Unbounded_operator" title="Unbounded operator">Un</a>)<a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Closed_linear_operator" title="Closed linear operator">Closed</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">on Hilbert spaces</a></li></ul></li> <li>(<a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Dis</a>)<a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Densely_defined" class="mw-redirect" title="Densely defined">Densely defined</a></li> <li>Fredholm <ul><li><a href="/wiki/Fredholm_kernel" title="Fredholm kernel">kernel</a></li> <li><a href="/wiki/Fredholm_operator" title="Fredholm operator">operator</a></li></ul></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Functionals</a> <ul><li><a href="/wiki/Positive_linear_functional" title="Positive linear functional">positive</a></li></ul></li> <li><a href="/wiki/Pseudo-monotone_operator" title="Pseudo-monotone operator">Pseudo-monotone</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Strictly_singular_operator" title="Strictly singular operator">Strictly singular</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a></li> <li><a href="/wiki/Operator_space" title="Operator space">Operator space</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a> <ul><li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">radius</a></li></ul></li> <li><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> <ul><li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">of ODEs</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li></ul></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_theorem" title="Banach–Mazur theorem">Banach–Mazur</a></li> <li><a href="/wiki/Banach%E2%80%93Saks_theorem" class="mw-redirect" title="Banach–Saks theorem">Banach–Saks</a></li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Banach–Schauder (open mapping)</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Banach–Steinhaus (Uniform boundedness)</a></li> <li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's inequality</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li> <li><a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">Closed graph</a></li> <li><a href="/wiki/Closed_range_theorem" title="Closed range theorem">Closed range</a></li> <li><a href="/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian</a></li> <li><a href="/wiki/Freudenthal_spectral_theorem" title="Freudenthal spectral theorem">Freudenthal spectral</a></li> <li><a href="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Goldstine_theorem" title="Goldstine theorem">Goldstine</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> <ul><li><a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li></ul></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Invariant_subspace_problem#Known_special_cases" title="Invariant subspace problem">Lomonosov's invariant subspace</a></li> <li><a href="/wiki/Mackey%E2%80%93Arens_theorem" title="Mackey–Arens theorem">Mackey–Arens</a></li> <li><a href="/wiki/Mazur%27s_lemma" title="Mazur's lemma">Mazur's lemma</a></li> <li><a href="/wiki/M._Riesz_extension_theorem" title="M. Riesz extension theorem">M. Riesz extension</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Riesz%27s_lemma" title="Riesz's lemma">Riesz's lemma</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson-Ursescu</a></li> <li><a href="/wiki/Schauder_fixed-point_theorem" title="Schauder fixed-point theorem">Schauder fixed-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analysis</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a></li> <li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a> <ul><li><a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Derivative" title="Derivative">Derivatives</a> <ul><li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux</a></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">functional</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">quasi</a></li></ul></li> <li><a href="/wiki/Integral" title="Integral">Integrals</a> <ul><li><a href="/wiki/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">weak</a></li></ul></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">holomorphic</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measures</a> <ul><li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Lebesgue</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_functions" class="mw-redirect" title="Strongly measurable functions">Strongly</a> measurable function</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced/Circled</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex</a></li> <li><a href="/wiki/Convex_cone" title="Convex cone">Convex cone (subset)</a></li> <li><a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))</li> <li><a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone (subset)</a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial</a></li> <li><a href="/wiki/Star_domain" title="Star domain">Radially convex/Star-shaped</a></li> <li><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a></li> <li><a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets / set operations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_hull" title="Affine hull">Affine hull</a></li> <li>(<a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li> <li><a href="/wiki/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li> <li><a href="/wiki/Polar_set" title="Polar set">Polar</a></li> <li>(<a href="/wiki/Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="/wiki/Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Template:ListOfBanachSpaces" class="mw-redirect" title="Template:ListOfBanachSpaces">Examples</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_continuity" title="Absolute continuity">Absolute continuity AC</a></li> <li><a href="/wiki/Ba_space" title="Ba space"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ba(\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>a</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ba(\Sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fe61351e3531b14043fa2d09e98c2437bd1a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle ba(\Sigma )}"></a></li> <li><a href="/wiki/C_space" title="C space">c space</a></li> <li><a href="/wiki/BK-space" title="BK-space">Banach coordinate BK</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p,q}^{s}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9919cf78ad095c237169772d2b27a37bfbef1b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.524ex; height:3.009ex;" alt="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"></a></li> <li><a href="/wiki/Birnbaum%E2%80%93Orlicz_space" class="mw-redirect" title="Birnbaum–Orlicz space">Birnbaum–Orlicz</a></li> <li><a href="/wiki/Bounded_variation" title="Bounded variation">Bounded variation BV</a></li> <li><a href="/wiki/Bs_space" title="Bs space">Bs space</a></li> <li><a href="/wiki/Continuous_functions_on_a_compact_Hausdorff_space" title="Continuous functions on a compact Hausdorff space">Continuous C(K) with K compact Hausdorff</a></li> <li><a href="/wiki/Hardy_space" title="Hardy space">Hardy Hp</a></li> <li><a class="mw-selflink-fragment" href="#Definition">Hilbert H</a></li> <li><a href="/wiki/Morrey%E2%80%93Campanato_space" title="Morrey–Campanato space">Morrey–Campanato <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\lambda ,p}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\lambda ,p}(\Omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8af58fa038369c3ec6386c6656aab82825e372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.545ex; height:3.176ex;" alt="{\displaystyle L^{\lambda ,p}(\Omega )}"></a></li> <li><a href="/wiki/Sequence_space#ℓp_spaces" title="Sequence space">ℓp</a> <ul><li><a href="/wiki/L-infinity#Sequence_space" title="L-infinity"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}"></a></li></ul></li> <li><a href="/wiki/Lp_space" title="Lp space">Lp</a> <ul><li><a href="/wiki/L-infinity#Function_space" title="L-infinity"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"></a></li> <li><a href="/wiki/Lp_space#Weighted_Lp_spaces" title="Lp space">weighted</a></li></ul></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow> <mo>(</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>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0465acd58a0f31e32b095aed742d9ccc6331369c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.592ex; height:2.843ex;" alt="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"></a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann F</a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence space</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev Wk,p</a> <ul><li><a href="/wiki/Sobolev_inequality" title="Sobolev inequality">Sobolev inequality</a></li></ul></li> <li><a href="/wiki/Triebel%E2%80%93Lizorkin_space" title="Triebel–Lizorkin space">Triebel–Lizorkin</a></li> <li><a href="/wiki/Wiener_amalgam_space" title="Wiener amalgam space">Wiener amalgam <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(X,L^{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(X,L^{p})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37b1dc9714960c525cb561a4828f41feb5844ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:2.843ex;" alt="{\displaystyle W(X,L^{p})}"></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Ordinary Differential Equations (ODEs)</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <a href="https://www.wikidata.org/wiki/Q190056#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4159850-7">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85060803">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11979628h">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11979628h">BnF data</a></li><li><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00563198">Japan</a></li><li><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">Czech Republic</a></li><li><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX531621">Spain</a></li><li><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007560453005171">Israel</a></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hilbert_space&oldid=1255844830">https://en.wikipedia.org/w/index.php?title=Hilbert_space&oldid=1255844830</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Hilbert_spaces" title="Category:Hilbert spaces">Hilbert spaces</a></li><li><a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Functional analysis</a></li><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Operator_theory" title="Category:Operator theory">Operator theory</a></li><li><a href="/wiki/Category:David_Hilbert" title="Category:David Hilbert">David Hilbert</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Pages_displaying_wikidata_descriptions_as_a_fallback_via_Module:Annotated_link" title="Category:Pages displaying wikidata descriptions as a fallback via Module:Annotated link">Pages displaying wikidata descriptions as a fallback via Module:Annotated link</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li><li><a href="/wiki/Category:Good_articles" title="Category:Good articles">Good articles</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"> This page was last edited on 6 November 2024, at 23:15 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Hilbert_space&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</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-f69cdc8f6-d8f47","wgBackendResponseTime":148,"wgPageParseReport":{"limitreport":{"cputime":"3.197","walltime":"3.837","ppvisitednodes":{"value":40881,"limit":1000000},"postexpandincludesize":{"value":387476,"limit":2097152},"templateargumentsize":{"value":58762,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":11,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":299506,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 3010.016 1 -total"," 25.57% 769.731 604 Template:Math"," 14.09% 424.164 77 Template:Citation"," 12.73% 383.066 2 Template:Reflist"," 10.81% 325.281 15 Template:Annotated_link"," 7.03% 211.572 6 Template:Navbox"," 6.67% 200.787 617 Template:Main_other"," 5.58% 167.845 1 Template:Short_description"," 5.51% 165.893 6 Template:Cite_book"," 5.13% 154.296 1 Template:Hilbert_space"]},"scribunto":{"limitreport-timeusage":{"value":"1.573","limit":"10.000"},"limitreport-memusage":{"value":19688266,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"Bessel\u0026#039;s_inequality\"] = 1,\n [\"CITEREFAxler2014\"] = 1,\n [\"CITEREFBachmanNariciBeckenstein2000\"] = 1,\n [\"CITEREFBersJohnSchechter1981\"] = 1,\n [\"CITEREFBerthier2020\"] = 1,\n [\"CITEREFBillingsley1986\"] = 1,\n [\"CITEREFBlanchetCharbit2014\"] = 1,\n [\"CITEREFBourbaki1986\"] = 1,\n [\"CITEREFBourbaki1987\"] = 1,\n [\"CITEREFBoyerMerzbach1991\"] = 1,\n [\"CITEREFBrennerScott2005\"] = 1,\n [\"CITEREFBrezis2010\"] = 1,\n [\"CITEREFButtazzoGiaquintaHildebrandt1998\"] = 1,\n [\"CITEREFClarkson1936\"] = 1,\n [\"CITEREFCourantHilbert1953\"] = 1,\n [\"CITEREFDieudonné1960\"] = 1,\n [\"CITEREFDirac1930\"] = 1,\n [\"CITEREFDunfordSchwartz1958\"] = 1,\n [\"CITEREFDuren1970\"] = 1,\n [\"CITEREFEinsiedlerWard2011\"] = 1,\n [\"CITEREFEvans1998\"] = 1,\n [\"CITEREFFolland1989\"] = 1,\n [\"CITEREFFolland2009\"] = 1,\n [\"CITEREFFréchet1904\"] = 1,\n [\"CITEREFFréchet1907\"] = 1,\n [\"CITEREFGiusti2003\"] = 1,\n [\"CITEREFGrattan-Guinness2000\"] = 1,\n [\"CITEREFHalmos1957\"] = 1,\n [\"CITEREFHalmos1982\"] = 1,\n [\"CITEREFHermann_Weyl2009\"] = 1,\n [\"CITEREFHewittStromberg1965\"] = 1,\n [\"CITEREFHilbertNordheimvon_Neumann1927\"] = 1,\n [\"CITEREFHolevo2001\"] = 1,\n [\"CITEREFKac1966\"] = 1,\n [\"CITEREFKadisonRingrose1983\"] = 1,\n [\"CITEREFKadisonRingrose1997\"] = 1,\n [\"CITEREFKakutani1939\"] = 1,\n [\"CITEREFKaratzasShreve2019\"] = 1,\n [\"CITEREFKline1972\"] = 1,\n [\"CITEREFKolmogorovFomin1970\"] = 1,\n [\"CITEREFKrantz2002\"] = 1,\n [\"CITEREFLanczos1988\"] = 1,\n [\"CITEREFLebesgue1904\"] = 1,\n [\"CITEREFLindenstraussTzafriri1971\"] = 1,\n [\"CITEREFMarsden1974\"] = 1,\n [\"CITEREFMurphy1990\"] = 1,\n [\"CITEREFNielsenChuang2000\"] = 1,\n [\"CITEREFPathria1996\"] = 1,\n [\"CITEREFPedersen1995\"] = 1,\n [\"CITEREFPeres1993\"] = 1,\n [\"CITEREFPrugovečki1981\"] = 1,\n [\"CITEREFPynchon1973\"] = 1,\n [\"CITEREFReedSimon1975\"] = 1,\n [\"CITEREFReedSimon1980\"] = 1,\n [\"CITEREFRieffelPolak2011\"] = 1,\n [\"CITEREFRiesz1907\"] = 1,\n [\"CITEREFRiesz1934\"] = 1,\n [\"CITEREFRieszSz.-Nagy1990\"] = 1,\n [\"CITEREFRoman2008\"] = 1,\n [\"CITEREFRudin1987\"] = 1,\n [\"CITEREFSaks2005\"] = 1,\n [\"CITEREFSchmidt1908\"] = 1,\n [\"CITEREFShubin1987\"] = 1,\n [\"CITEREFSobrino1996\"] = 1,\n [\"CITEREFStapleton1995\"] = 1,\n [\"CITEREFStein1970\"] = 1,\n [\"CITEREFSteinShakarchi2005\"] = 1,\n [\"CITEREFSteinWeiss1971\"] = 1,\n [\"CITEREFStewart2006\"] = 1,\n [\"CITEREFStreaterWightman1964\"] = 1,\n [\"CITEREFStroock2011\"] = 1,\n [\"CITEREFTeschl2009\"] = 1,\n [\"CITEREFTitchmarsh1946\"] = 1,\n [\"CITEREFTrèves1967\"] = 1,\n [\"CITEREFWarner1983\"] = 1,\n [\"CITEREFWeidmann1980\"] = 1,\n [\"CITEREFWeyl1931\"] = 1,\n [\"CITEREFYoung1988\"] = 1,\n [\"CITEREFvon_Neumann1929\"] = 1,\n [\"CITEREFvon_Neumann1932\"] = 1,\n [\"CITEREFvon_Neumann1955\"] = 1,\n [\"Parseval\u0026#039;s_formula\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 2,\n [\"Abramowitz_Stegun_ref\"] = 1,\n [\"Abs\"] = 2,\n [\"Anchor\"] = 2,\n [\"Annotated link\"] = 15,\n [\"Authority control\"] = 1,\n [\"Banach spaces\"] = 1,\n [\"Bra\"] = 1,\n [\"Bra-ket\"] = 2,\n [\"Citation\"] = 77,\n [\"Cite book\"] = 3,\n [\"Closed-closed\"] = 1,\n [\"Colend\"] = 1,\n [\"Cols\"] = 1,\n [\"Commons category\"] = 1,\n [\"Em\"] = 9,\n [\"For\"] = 1,\n [\"Functional analysis\"] = 1,\n [\"Good article\"] = 1,\n [\"Harv\"] = 1,\n [\"Harvnb\"] = 90,\n [\"Harvtxt\"] = 35,\n [\"Hilbert space\"] = 1,\n [\"I sup\"] = 2,\n [\"Ket\"] = 1,\n [\"Lp spaces\"] = 1,\n [\"MacTutor\"] = 1,\n [\"Main\"] = 7,\n [\"Math\"] = 603,\n [\"Math theorem\"] = 2,\n [\"Mvar\"] = 24,\n [\"Norm\"] = 7,\n [\"Nowrap\"] = 2,\n [\"Portal\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 2,\n [\"Rudin Walter Functional Analysis\"] = 1,\n [\"Schaefer Wolff Topological Vector Spaces\"] = 1,\n [\"Sfn\"] = 1,\n [\"Sfrac\"] = 4,\n [\"Short description\"] = 1,\n [\"Springer\"] = 2,\n [\"Su\"] = 10,\n [\"Wikibooks\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n","limitreport-profile":[["?","280","16.5"],["recursiveClone \u003CmwInit.lua:45\u003E","240","14.1"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getAllExpandedArguments","200","11.8"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getExpandedArgument","160","9.4"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","160","9.4"],["dataWrapper \u003Cmw.lua:672\u003E","100","5.9"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::gsub","80","4.7"],["select_one \u003CModule:Citation/CS1/Utilities:426\u003E","60","3.5"],["type","40","2.4"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::sub","40","2.4"],["[others]","340","20.0"]]},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-lsrvg","timestamp":"20241124053517","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Hilbert space","url":"https:\/\/en.wikipedia.org\/wiki\/Hilbert_space","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":"2001-11-02T21:29:08Z","dateModified":"2024-11-06T23:15:07Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/5\/5c\/Standing_waves_on_a_string.gif","headline":"inner product space that is metrically complete; a Banach space whose norm induces an inner product (follows the parallelogram identity)"}</script> </body> </html>