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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Definicija_i_ilustracija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicija_i_ilustracija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicija i ilustracija</span> </div> </a> <button aria-controls="toc-Definicija_i_ilustracija-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Uključi/isključi podsekciju Definicija i ilustracija</span> </button> <ul id="toc-Definicija_i_ilustracija-sublist" class="vector-toc-list"> <li id="toc-Uvodni_primer:_euklidski_prostor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uvodni_primer:_euklidski_prostor"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Uvodni primer: euklidski prostor</span> </div> </a> <ul id="toc-Uvodni_primer:_euklidski_prostor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definicija" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definicija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definicija</span> </div> </a> <ul id="toc-Definicija-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Reference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spoljašnje_veze" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spoljašnje_veze"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Spoljašnje veze</span> </div> </a> <ul id="toc-Spoljašnje_veze-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="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" lang="sh-Latn" dir="ltr"> <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="Prikaži/sakrij sadržaj" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Prikaži/sakrij sadržaj</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading" lang="sh-Latn" dir="ltr"><span class="mw-page-title-main">Hilbertov prostor</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" lang="sh-Latn" dir="ltr"> <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="Prijedi na druge jezične varijante članka. Dostupno je na 59 jezika" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-59" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">59 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hilbert-ruimte" title="Hilbert-ruimte — Afrikanerski" lang="af" hreflang="af" data-title="Hilbert-ruimte" data-language-autonym="Afrikaans" data-language-local-name="Afrikanerski" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D9%87%D9%8A%D9%84%D8%A8%D8%B1%D8%AA" title="فضاء هيلبرت — Arapski" lang="ar" hreflang="ar" data-title="فضاء هيلبرت" data-language-autonym="العربية" data-language-local-name="Arapski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_de_Hilbert" title="Espaciu de Hilbert — Asturijski" lang="ast" hreflang="ast" data-title="Espaciu de Hilbert" data-language-autonym="Asturianu" data-language-local-name="Asturijski" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Hilbert_f%C9%99zas%C4%B1" title="Hilbert fəzası — Azerbejdžanski" lang="az" hreflang="az" data-title="Hilbert fəzası" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbejdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82_%D0%B0%D1%80%D0%B0%D1%83%D1%8B%D2%93%D1%8B" title="Гильберт арауығы — Baškir" lang="ba" hreflang="ba" data-title="Гильберт арауығы" data-language-autonym="Башҡортса" data-language-local-name="Baškir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BB%D0%B1%D0%B5%D1%80%D1%82%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Хилбертово пространство — Bugarski" lang="bg" hreflang="bg" data-title="Хилбертово пространство" data-language-autonym="Български" data-language-local-name="Bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A6%BF%E0%A6%B2%E0%A6%AC%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%9F_%E0%A6%9C%E0%A6%97%E0%A7%8E" title="হিলবার্ট জগৎ — Bengalski" lang="bn" hreflang="bn" data-title="হিলবার্ট জগৎ" data-language-autonym="বাংলা" data-language-local-name="Bengalski" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_de_Hilbert" title="Espai de Hilbert — Katalonski" lang="ca" hreflang="ca" data-title="Espai de Hilbert" data-language-autonym="Català" data-language-local-name="Katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%DA%BE%DB%8C%D9%84%D8%A8%DB%8E%D8%B1%D8%AA" title="بۆشاییی ھیلبێرت — centralnokurdski" lang="ckb" hreflang="ckb" data-title="بۆشاییی ھیلبێرت" data-language-autonym="کوردی" data-language-local-name="centralnokurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hilbert%C5%AFv_prostor" title="Hilbertův prostor — Češki" lang="cs" hreflang="cs" data-title="Hilbertův prostor" data-language-autonym="Čeština" data-language-local-name="Češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hilbertrum" title="Hilbertrum — Danski" lang="da" hreflang="da" data-title="Hilbertrum" data-language-autonym="Dansk" data-language-local-name="Danski" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hilbertraum" title="Hilbertraum — Nemački" lang="de" hreflang="de" data-title="Hilbertraum" data-language-autonym="Deutsch" data-language-local-name="Nemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A7%CF%8E%CF%81%CE%BF%CF%82_%CE%A7%CE%AF%CE%BB%CE%BC%CF%80%CE%B5%CF%81%CF%84" title="Χώρος Χίλμπερτ — Grčki" lang="el" hreflang="el" data-title="Χώρος Χίλμπερτ" data-language-autonym="Ελληνικά" data-language-local-name="Grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="dobar članak"><a href="https://en.wikipedia.org/wiki/Hilbert_space" title="Hilbert space — Engleski" lang="en" hreflang="en" data-title="Hilbert space" data-language-autonym="English" data-language-local-name="Engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_de_Hilbert" title="Espacio de Hilbert — Španski" lang="es" hreflang="es" data-title="Espacio de Hilbert" data-language-autonym="Español" data-language-local-name="Španski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hilberti_ruum" title="Hilberti ruum — Estonski" lang="et" hreflang="et" data-title="Hilberti ruum" data-language-autonym="Eesti" data-language-local-name="Estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Hilberten_espazio" title="Hilberten espazio — Baskijski" lang="eu" hreflang="eu" data-title="Hilberten espazio" data-language-autonym="Euskara" data-language-local-name="Baskijski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%87%DB%8C%D9%84%D8%A8%D8%B1%D8%AA" title="فضای هیلبرت — Persijski" lang="fa" hreflang="fa" data-title="فضای هیلبرت" data-language-autonym="فارسی" data-language-local-name="Persijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hilbertin_avaruus" title="Hilbertin avaruus — Finski" lang="fi" hreflang="fi" data-title="Hilbertin avaruus" data-language-autonym="Suomi" data-language-local-name="Finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_de_Hilbert" title="Espace de Hilbert — Francuski" lang="fr" hreflang="fr" data-title="Espace de Hilbert" data-language-autonym="Français" data-language-local-name="Francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_de_Hilbert" title="Espazo de Hilbert — Galski" lang="gl" hreflang="gl" data-title="Espazo de Hilbert" data-language-autonym="Galego" data-language-local-name="Galski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%94%D7%99%D7%9C%D7%91%D7%A8%D7%98" title="מרחב הילברט — Hebrejski" lang="he" hreflang="he" data-title="מרחב הילברט" data-language-autonym="עברית" data-language-local-name="Hebrejski" class="interlanguage-link-target"><span>עברית</span></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 — Mađarski" lang="hu" hreflang="hu" data-title="Hilbert-tér" data-language-autonym="Magyar" data-language-local-name="Mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%AC%D5%A2%D5%A5%D6%80%D5%BF%D5%B5%D5%A1%D5%B6_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հիլբերտյան տարածություն — Jermenski" lang="hy" hreflang="hy" data-title="Հիլբերտյան տարածություն" data-language-autonym="Հայերեն" data-language-local-name="Jermenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_Hilbert" title="Ruang Hilbert — Indonezijski" lang="id" hreflang="id" data-title="Ruang Hilbert" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hilbert-r%C3%BAm" title="Hilbert-rúm — Islandski" lang="is" hreflang="is" data-title="Hilbert-rúm" data-language-autonym="Íslenska" data-language-local-name="Islandski" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert — Italijanski" lang="it" hreflang="it" data-title="Spazio di Hilbert" data-language-autonym="Italiano" data-language-local-name="Italijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%92%E3%83%AB%E3%83%99%E3%83%AB%E3%83%88%E7%A9%BA%E9%96%93" title="ヒルベルト空間 — Japanski" lang="ja" hreflang="ja" data-title="ヒルベルト空間" data-language-autonym="日本語" data-language-local-name="Japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9E%90%EB%B2%A0%EB%A5%B4%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="힐베르트 공간 — Korejski" lang="ko" hreflang="ko" data-title="힐베르트 공간" data-language-autonym="한국어" data-language-local-name="Korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82_%D0%BC%D0%B5%D0%B9%D0%BA%D0%B8%D0%BD%D0%B4%D0%B8%D0%B3%D0%B8" title="Гильберт мейкиндиги — Kirgiski" lang="ky" hreflang="ky" data-title="Гильберт мейкиндиги" data-language-autonym="Кыргызча" data-language-local-name="Kirgiski" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Hilberto_erdv%C4%97" title="Hilberto erdvė — Litvanski" lang="lt" hreflang="lt" data-title="Hilberto erdvė" data-language-autonym="Lietuvių" data-language-local-name="Litvanski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BB%D0%B1%D0%B5%D1%80%D1%82%D0%B8%D0%B9%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD_%D0%B7%D0%B0%D0%B9" title="Хилбертийн орон зай — Mongolski" lang="mn" hreflang="mn" data-title="Хилбертийн орон зай" data-language-autonym="Монгол" data-language-local-name="Mongolski" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_Hilbert" title="Ruang Hilbert — Malajski" lang="ms" hreflang="ms" data-title="Ruang Hilbert" data-language-autonym="Bahasa Melayu" data-language-local-name="Malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hilbertruimte" title="Hilbertruimte — Holandski" lang="nl" hreflang="nl" data-title="Hilbertruimte" data-language-autonym="Nederlands" data-language-local-name="Holandski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hilbertrom" title="Hilbertrom — Norveški njorsk" lang="nn" hreflang="nn" data-title="Hilbertrom" data-language-autonym="Norsk nynorsk" data-language-local-name="Norveški njorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hilbert-rom" title="Hilbert-rom — Norveški bokmål" lang="nb" hreflang="nb" data-title="Hilbert-rom" data-language-autonym="Norsk bokmål" data-language-local-name="Norveški bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B9%E0%A8%BF%E0%A8%B2%E0%A8%AC%E0%A8%B0%E0%A8%9F_%E0%A8%B8%E0%A8%AA%E0%A9%87%E0%A8%B8" title="ਹਿਲਬਰਟ ਸਪੇਸ — Pandžabski" lang="pa" hreflang="pa" data-title="ਹਿਲਬਰਟ ਸਪੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Pandžabski" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_Hilberta" title="Przestrzeń Hilberta — Poljski" lang="pl" hreflang="pl" data-title="Przestrzeń Hilberta" data-language-autonym="Polski" data-language-local-name="Poljski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DB%81%D9%84%D8%A8%D8%B1%D9%B9_%D8%B3%D9%BE%DB%8C%D8%B3" title="ہلبرٹ سپیس — Western Punjabi" lang="pnb" hreflang="pnb" data-title="ہلبرٹ سپیس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_de_Hilbert" title="Espaço de Hilbert — Portugalski" lang="pt" hreflang="pt" data-title="Espaço de Hilbert" data-language-autonym="Português" data-language-local-name="Portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_Hilbert" title="Spațiu Hilbert — Rumunski" lang="ro" hreflang="ro" data-title="Spațiu Hilbert" data-language-autonym="Română" data-language-local-name="Rumunski" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Гильбертово пространство — Ruski" lang="ru" hreflang="ru" data-title="Гильбертово пространство" data-language-autonym="Русский" data-language-local-name="Ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Hilbert_space" title="Hilbert space — Škotski" lang="sco" hreflang="sco" data-title="Hilbert space" data-language-autonym="Scots" data-language-local-name="Škotski" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hilbert_space" title="Hilbert space — Simple English" lang="en-simple" hreflang="en-simple" data-title="Hilbert space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hilbertov_priestor" title="Hilbertov priestor — Slovački" lang="sk" hreflang="sk" data-title="Hilbertov priestor" data-language-autonym="Slovenčina" data-language-local-name="Slovački" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hilbertov_prostor" title="Hilbertov prostor — Slovenački" lang="sl" hreflang="sl" data-title="Hilbertov prostor" data-language-autonym="Slovenščina" data-language-local-name="Slovenački" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a 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class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espasyong_Hilbert" title="Espasyong Hilbert — Tagalski" lang="tl" hreflang="tl" data-title="Espasyong Hilbert" data-language-autonym="Tagalog" data-language-local-name="Tagalski" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hilbert_uzay%C4%B1" title="Hilbert uzayı — Turski" lang="tr" hreflang="tr" data-title="Hilbert uzayı" data-language-autonym="Türkçe" data-language-local-name="Turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%96%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82%D1%96%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Гільбертів простір — Ukrajinski" lang="uk" hreflang="uk" 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class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q190056" title="Pogledajte povezanu stavku u skladištu podataka [g]" accesskey="g"><span>Stavka na Wikipodacima</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Pomagala"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Izgled"> <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">Izgled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">premjesti na bočnu traku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sakrij</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Izvor: Wikipedija</div> </div> <div id="contentSub" lang="sh-Latn" dir="ltr"><div id="mw-content-subtitle" lang="sh-Latn" dir="ltr"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sh-Latn" dir="ltr"><p><b>Hilbertov prostor</b> je <a href="/wiki/Matematika" title="Matematika">matematički koncept</a> koji generalizuje <a href="/wiki/Euklidski_prostor" title="Euklidski prostor">euklidski prostor</a>. U njemu se metode <a href="/wiki/Linearna_algebra" title="Linearna algebra">vektorske algebre</a> i <a href="/wiki/Matemati%C4%8Dka_analiza" title="Matematička analiza">analize</a> iz euklidske ravni i trodimenzionalnog prostora proširuju na prostor sa konačnim ili beskonačnim brojem <a href="/wiki/Dimenzija" title="Dimenzija">dimenzija</a>. Dobio je ime po <a href="/wiki/David_Hilbert" title="David Hilbert">Davidu Hilbertu</a>. </p><p>Hilbertov prostor se često pojavljuje u <a href="/wiki/Matematika" title="Matematika">matematici</a>, <a href="/wiki/Fizika" title="Fizika">fizici</a> i <a href="/wiki/In%C5%BEinjerstvo" class="mw-redirect" title="Inžinjerstvo">inžinjerstvu</a>, tipično kao preslikavanja beskonačnog broja dimenzija. </p><p>Geometrijske analogije imaju veliki značaj u razumevanju teorije Hilbertovih prostora. Za njih postoji ekvivalentna <a href="/wiki/Pitagorina_teorema" title="Pitagorina teorema">Pitagorina teorema</a> i <a href="/w/index.php?title=Zakon_paralelograma&action=edit&redlink=1" class="new" title="Zakon paralelograma (stranica ne postoji)">zakon paralelograma</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicija_i_ilustracija">Definicija i ilustracija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbertov_prostor&veaction=edit&section=1" title="Uredi odjeljak Definicija i ilustracija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hilbertov_prostor&action=edit&section=1" title="Uredi kôd odjeljka Definicija i ilustracija"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Uvodni_primer:_euklidski_prostor">Uvodni primer: euklidski prostor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbertov_prostor&veaction=edit&section=2" title="Uredi odjeljak Uvodni primer: euklidski prostor" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hilbertov_prostor&action=edit&section=2" title="Uredi kôd odjeljka Uvodni primer: euklidski prostor"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jedan od najjasnijih primera Hilbertovih prostora je <a href="/wiki/Euklidski_prostor" title="Euklidski prostor">euklidski prostor</a> koji se sastoji iz trodimenzionalnih vektora, označavamo ga sa <b>R</b><sup>3</sup>, u kome je definisan operator proizvoda. Ovaj proizvod uzima dva vektora <b>x</b> i <b>y</b> kao argumente i kao rezultat daje realan broj <b>x</b>·<b>y</b>. Ako su <b>x</b> i <b>y</b> predstavljeni u <a href="/wiki/Dekartov_koordinatni_sistem" class="mw-redirect" title="Dekartov koordinatni sistem">Dekartovim koordinatama</a>, onda se operator proizvoda definiše kao: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},x_{3})\cdot (y_{1},y_{2},y_{3})=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</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>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},x_{3})\cdot (y_{1},y_{2},y_{3})=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b045b6e2aaa82623b68284cc3e7d797207a61df5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.324ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},x_{3})\cdot (y_{1},y_{2},y_{3})=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}"></span></dd></dl> <p>Operator proizvoda zadovoljava sledeće uslove: </p> <ol><li>Simetričan je u odnosu na <b>x</b> i <b>y</b>: <b>x</b>·<b>y</b> = <b>y</b>·<b>x</b>.</li> <li>Linearan je u odnosu na prvi argument: (<i>a</i><b>x</b><sub>1</sub> + <i>b</i><b>x</b><sub>2</sub>)·<b>y</b> = <i>a</i><b>x</b><sub>1</sub>·<b>y</b> + <i>b</i><b>x</b><sub>2</sub>·<b>y</b> za bilo koje skalare <i>a</i>, <i>b</i> i vektore <b>x</b><sub>1</sub>, <b>x</b><sub>2</sub> i <b>y</b>.</li> <li>To je pozitivna <a href="/w/index.php?title=Bilinearna_forma&action=edit&redlink=1" class="new" title="Bilinearna forma (stranica ne postoji)">bilinearna forma</a>: za sve vektore <b>x</b>, <b>x</b>·<b>x</b> ≥ 0, gde znak jednakosti važi <a href="/wiki/Ako_i_samo_ako" title="Ako i samo ako">ako i samo ako</a> je <b>x</b> = 0.</li></ol> <p>Operacija nad parom vektora koja zadovoljava ova tri uslova se naziva <a href="/w/index.php?title=Vektorski_proizvod&action=edit&redlink=1" class="new" title="Vektorski proizvod (stranica ne postoji)">vektorski proizvod</a>. Svaki vektorski prostor sa konačnim brojem dimenzija u kome je definisan vektorski proizvod predstavlja Hilbertov prostor. Karakteristika goredefinisanog operatora množenja koja ga povezuje sa euklidskom geometrijom je što zavisi i od dužine (ili intenziteta) vektora, koji se označava sa ||<b>x</b>||, i od ugla θ između vektora <b>x</b> i <b>y</b>. Ta zavisnost se izražava formulom: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\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> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\cos \theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5422e6f5681449549b32a28c8e00116af8a2c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.468ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\cos \theta .}"></span></dd></dl> <p><a href="/w/index.php?title=Matemati%C4%8Dki_niz&action=edit&redlink=1" class="new" title="Matematički niz (stranica ne postoji)">Matematički niz</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3d791a8b023df5a980200e276eec54d69eb2b6" class="mwe-math-fallback-image-inline 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}}"></span></dd></dl> <p>koji se sastoji iz vektora u <b>R</b><sup>3</sup> <a href="/w/index.php?title=Apsolutna_konvergencija&action=edit&redlink=1" class="new" title="Apsolutna konvergencija (stranica ne postoji)">apsolutno konvergira</a> pod uslovom da suma dužina konvergira (kao u slučaju niza realnih brojeva): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a887eb04138b4edfa006f4e8aa97b259b42e51f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.636ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty .}"></span></dd></dl> <p>Slično nizu skalara, niz vektora koji apsolutno konvergira istovremeno konvergira ka nekom vektoru <b>L</b> u euklidskom prostoru, i to tako da: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{kada }}N\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <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> <mo symmetric="true">‖</mo> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>kada </mtext> </mrow> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{kada }}N\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d4821a89d2475bd1b4aecec2e89ca84fe61c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.188ex; height:7.509ex;" alt="{\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{kada }}N\to \infty .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Definicija">Definicija</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbertov_prostor&veaction=edit&section=3" title="Uredi odjeljak Definicija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hilbertov_prostor&action=edit&section=3" title="Uredi kôd odjeljka Definicija"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hilbertov prostor <i>H</i> je <a href="/wiki/Realni_brojevi" class="mw-redirect" title="Realni brojevi">realan</a> ili <a href="/wiki/Kompleksni_brojevi" class="mw-redirect" title="Kompleksni brojevi">kompleksan</a> <a href="/wiki/Vektorski_prostor" title="Vektorski prostor">vektorski prostor</a> koji je istovremeno i <a href="/w/index.php?title=Ko%C5%A1ijev_metri%C4%8Dki_prostor&action=edit&redlink=1" class="new" title="Košijev metrički prostor (stranica ne postoji)">Košijev metrički prostor</a> u odnosu na metričku funkciju vektorskog proizvoda. Kakda kažemo da je <i>H</i> kompleksni vektorski prostor, to znači da u <i>H</i> postoji proizvod ⟨<i>x</i>,<i>y</i>⟩ koji paru elemenata <i>x</i>,<i>y'- iz </i>H<i>, pridružuje kompleksnu vrednost, pri čemu je:</i> </p> <ul><li>⟨<i>y</i>,<i>x</i>⟩ je <a href="/w/index.php?title=Konjugovan_kompleksan_broj&action=edit&redlink=1" class="new" title="Konjugovan kompleksan broj (stranica ne postoji)">konjugovan kompleksan broj</a> od ⟨<i>x</i>,<i>y</i>⟩:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/530e809e1c223b58512514a0eb54e4ad4f0072b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.517ex; height:3.676ex;" alt="{\displaystyle \langle y,x\rangle ={\overline {\langle x,y\rangle }}.}"></span></dd></dl></dd></dl> <ul><li>⟨<i>x</i>,<i>y</i>⟩ je linearna po prvom argumentu. Za sve kompleksne brojeve <i>a</i> i <i>b</i>,</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle 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> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1dc80b4950257247bef594013fe77175bc9f784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.413ex; height:2.843ex;" alt="{\displaystyle \langle ax_{1}+bx_{2},y\rangle =a\langle x_{1},y\rangle +b\langle x_{2},y\rangle .}"></span></dd></dl></dd></dl> <ul><li>Proizvod je pozitivna <a href="/w/index.php?title=Bilinearna_forma&action=edit&redlink=1" class="new" title="Bilinearna forma (stranica ne postoji)">bilinearna forma</a>:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle \geq 0}"> <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> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6bce7d4d0f12edae272a1986cb3d93a305bdcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle \geq 0}"></span></dd></dl></dd> <dd>gde znak jednakosti važi za <i>x</i> = 0.</dd></dl> <p>Realni vektorski prostor se definiše na isti način, osim što vektorski proizvod ima realne vrednosti. </p><p><a href="/w/index.php?title=Intenzitet_vektora&action=edit&redlink=1" class="new" title="Intenzitet vektora (stranica ne postoji)">Intenzitet vektora</a> definiše se kao proizvod ⟨•,•⟩ u obliku realne funkcije: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77344e63ba35620dfd6c32794ede2e704fb78ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.226ex; height:4.843ex;" alt="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }},}"></span></dd></dl> <p>a rastojanje između tačaka <i>x</i>,<i>y</i> u <i>H</i> definiše se pomoću intenziteta na sledeći način: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=\|x-y\|={\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> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d573029a8b67be2ce083ad4452fcc215b9f4d66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.857ex; height:4.843ex;" alt="{\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}.}"></span></dd></dl> <p>Ovo je funkcija metrike, što znači da (1) da je simetrična po <i>x</i> i <i>y</i>, (2) da je rastojanje između <i>x</i> i <i>x</i> nula, a da su ostala rastojanja između <i>x</i> i <i>y</i> pozitivna, (3) da važi <a href="/w/index.php?title=Nejednakost_trougla&action=edit&redlink=1" class="new" title="Nejednakost trougla (stranica ne postoji)">nejednakost trougla</a>, što znači da dužina stranice a u trouglu <i>xyz</i> ne može biti duža od zbira preostale dve stranice: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7072502a79725a25151d5c96b7310e7dbb97a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.91ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}"></span></dd></dl> <p>Poslednja osobina je posledica <a href="/w/index.php?title=Ko%C5%A1i-%C5%A0varcova_nejednakost&action=edit&redlink=1" class="new" title="Koši-Švarcova nejednakost (stranica ne postoji)">Koši-Švarcove nejednakosti</a> koja tvrdi da: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f146a809cdbeebe4de138cc45d3073b26e6e9312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.243ex; height:2.843ex;" alt="{\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}"></span></dd></dl> <p>gde znak jednakosti važi kada su <i>x</i> i <i>y</i> <a href="/w/index.php?title=Linearna_nezavisnost&action=edit&redlink=1" class="new" title="Linearna nezavisnost (stranica ne postoji)">linearno zavisni</a>. </p><p>Kada se funkcija udaljenosti definiše na ovaj način, kao <a href="/w/index.php?title=Funkcija_metrike&action=edit&redlink=1" class="new" title="Funkcija metrike (stranica ne postoji)">funkcija metrike</a>, onda vektorski prostor postaje <b>pre-Hilbertov prostor</b>. Svaki kopmpletan pre-Hilbertov prostor je Hilbertov prostor. Kompletnost se definiše uslovom: ako za niz vektora važi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\sum _{k=0}^{\infty }u_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\sum _{k=0}^{\infty }u_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed719d23cc0abf0ff9388095aef7564066caa041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.449ex; height:3.176ex;" alt="{\displaystyle \textstyle {\sum _{k=0}^{\infty }u_{k}}}"></span> <a href="/w/index.php?title=Apsolutna_konvergencija&action=edit&redlink=1" class="new" title="Apsolutna konvergencija (stranica ne postoji)">apsolutno konvergira</a> tako da </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01cc6959b59fe6c87654db8ca38118ee8c8fa1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.554ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty ,}"></span></dd></dl> <p>tada niz konvergira u <i>H</i>, u smislu da parcijalne sume teže nekom elementu <i>H</i>. </p><p>Kao Košijevi normirani prostori, Hilbertovi prostori su po definiciji i <a href="/w/index.php?title=Banahov_prostor&action=edit&redlink=1" class="new" title="Banahov prostor (stranica ne postoji)">Banahovi prostori</a>. Oni su i <a href="/w/index.php?title=Topolo%C5%A1ki_vektorski_prostori&action=edit&redlink=1" class="new" title="Topološki vektorski prostori (stranica ne postoji)">topološki vektorski prostori</a> u kojima su definisaani <a href="/w/index.php?title=Topologija_(matematika)&action=edit&redlink=1" class="new" title="Topologija (matematika) (stranica ne postoji)">topološki</a> pojmovi otvorenih i zatvorenih podskupova. </p> <div class="mw-heading mw-heading2"><h2 id="Reference">Reference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbertov_prostor&veaction=edit&section=4" title="Uredi odjeljak Reference" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hilbertov_prostor&action=edit&section=4" title="Uredi kôd odjeljka Reference"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r41556849">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.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> <div class="mw-heading mw-heading2"><h2 id="Spoljašnje_veze"><span id="Spolja.C5.A1nje_veze"></span>Spoljašnje veze</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hilbertov_prostor&veaction=edit&section=5" title="Uredi odjeljak Spoljašnje veze" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hilbertov_prostor&action=edit&section=5" title="Uredi kôd odjeljka Spoljašnje veze"><span>uredi kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/HilbertSpace.html">Hilbertov prostor na Mathworld</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="" lang="sh-Latn" dir="ltr">Izvor: <a dir="ltr" href="https://sh.wikipedia.org/w/index.php?title=Hilbertov_prostor&oldid=40755808">https://sh.wikipedia.org/w/index.php?title=Hilbertov_prostor&oldid=40755808</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Posebno:Kategorije" title="Posebno:Kategorije">Kategorije</a>: <ul><li><a href="/wiki/Kategorija:Linearna_algebra" title="Kategorija:Linearna algebra">Linearna algebra</a></li><li><a href="/wiki/Kategorija:Kvantna_mehanika" title="Kategorija:Kvantna mehanika">Kvantna mehanika</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" lang="sh-Latn" dir="ltr"> <ul id="footer-info"> <li id="footer-info-lastmod"> Ova stranica je posljednji put izmijenjena 28. juna 2017. u 20:46.</li> <li id="footer-info-copyright">Tekst je dostupan pod licencom <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-ShareAlike</a>; mogu se primijeniti i dodatni uslovi. 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