Teorema di approssimazione di Weierstrass

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class="vector-toc-link" href="#Enunciato"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Enunciato</span> </div> </a> <ul id="toc-Enunciato-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimostrazione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dimostrazione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Dimostrazione</span> </div> </a> <button aria-controls="toc-Dimostrazione-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>Attiva/disattiva la sottosezione Dimostrazione</span> </button> <ul id="toc-Dimostrazione-sublist" class="vector-toc-list"> <li id="toc-Osservazioni_preliminari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Osservazioni_preliminari"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Osservazioni preliminari</span> </div> </a> <ul id="toc-Osservazioni_preliminari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definizione_e_proprietà_dei_polinomi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definizione_e_proprietà_dei_polinomi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Definizione e proprietà dei polinomi</span> </div> </a> <ul id="toc-Definizione_e_proprietà_dei_polinomi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parte_principale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parte_principale"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Parte principale</span> </div> </a> <ul id="toc-Parte_principale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Caso_complesso" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Caso_complesso"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Caso complesso</span> </div> </a> <ul id="toc-Caso_complesso-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enunciato_del_teorema_tramite_i_concetti_degli_spazi_normati" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enunciato_del_teorema_tramite_i_concetti_degli_spazi_normati"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Enunciato del teorema tramite i concetti degli spazi normati</span> </div> </a> <ul id="toc-Enunciato_del_teorema_tramite_i_concetti_degli_spazi_normati-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conseguenze" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conseguenze"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Conseguenze</span> </div> </a> <button aria-controls="toc-Conseguenze-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>Attiva/disattiva la sottosezione Conseguenze</span> </button> <ul id="toc-Conseguenze-sublist" class="vector-toc-list"> <li id="toc-Risvolti_teorici" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Risvolti_teorici"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Risvolti teorici</span> </div> </a> <ul id="toc-Risvolti_teorici-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Risvolti_pratici" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Risvolti_pratici"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Risvolti pratici</span> </div> </a> <ul id="toc-Risvolti_pratici-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Il_teorema_di_Stone-Weierstrass" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Il_teorema_di_Stone-Weierstrass"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Il teorema di Stone-Weierstrass</span> </div> </a> <ul id="toc-Il_teorema_di_Stone-Weierstrass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ulteriori_generalizzazioni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ulteriori_generalizzazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Ulteriori generalizzazioni</span> </div> </a> <button aria-controls="toc-Ulteriori_generalizzazioni-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>Attiva/disattiva la sottosezione Ulteriori generalizzazioni</span> </button> <ul id="toc-Ulteriori_generalizzazioni-sublist" class="vector-toc-list"> <li id="toc-Teorema_di_Stone-Weierstrass_per_i_reticoli_di_funzioni_continue" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_di_Stone-Weierstrass_per_i_reticoli_di_funzioni_continue"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Teorema di Stone-Weierstrass per i reticoli di funzioni continue</span> </div> </a> <ul id="toc-Teorema_di_Stone-Weierstrass_per_i_reticoli_di_funzioni_continue-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_di_Bishop" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_di_Bishop"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Teorema di Bishop</span> </div> </a> <ul id="toc-Teorema_di_Bishop-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Teorema di approssimazione di Weierstrass</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 17 lingue" > <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-17" 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">17 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%B3%D8%AA%D9%88%D9%86-%D9%81%D8%A7%D9%8A%D8%B1%D8%B4%D8%AA%D8%B1%D8%A7%D8%B3" title="مبرهنة ستون-فايرشتراس - arabo" lang="ar" hreflang="ar" data-title="مبرهنة ستون-فايرشتراس" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_d%27Stone-Weierstrass" title="Teorema d'Stone-Weierstrass - catalano" lang="ca" hreflang="ca" data-title="Teorema d'Stone-Weierstrass" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Stone-Weierstra%C3%9F" title="Satz von Stone-Weierstraß - tedesco" lang="de" hreflang="de" data-title="Satz von Stone-Weierstraß" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem" title="Stone–Weierstrass theorem - inglese" lang="en" hreflang="en" data-title="Stone–Weierstrass theorem" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_aproximaci%C3%B3n_de_Weierstrass" title="Teorema de aproximación de Weierstrass - spagnolo" lang="es" hreflang="es" data-title="Teorema de aproximación de Weierstrass" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Stone-Weierstrass" title="Théorème de Stone-Weierstrass - francese" lang="fr" hreflang="fr" data-title="Théorème de Stone-Weierstrass" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A7%D7%99%D7%A8%D7%95%D7%91_%D7%A9%D7%9C_%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפט הקירוב של ויירשטראס - ebraico" lang="he" hreflang="he" data-title="משפט הקירוב של ויירשטראס" data-language-autonym="עברית" data-language-local-name="ebraico" 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/Weierstrass_approxim%C3%A1ci%C3%B3s_t%C3%A9tele" title="Weierstrass approximációs tétele - ungherese" lang="hu" hreflang="hu" data-title="Weierstrass approximációs tétele" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B9%E3%83%88%E3%83%BC%E3%83%B3%EF%BC%9D%E3%83%AF%E3%82%A4%E3%82%A8%E3%83%AB%E3%82%B7%E3%83%A5%E3%83%88%E3%83%A9%E3%82%B9%E3%81%AE%E5%AE%9A%E7%90%86" title="ストーン＝ワイエルシュトラスの定理 - giapponese" lang="ja" hreflang="ja" data-title="ストーン＝ワイエルシュトラスの定理" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Stone-Weierstrass" title="Stelling van Stone-Weierstrass - olandese" lang="nl" hreflang="nl" data-title="Stelling van Stone-Weierstrass" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Stone%E2%80%99a-Weierstrassa" title="Twierdzenie Stone’a-Weierstrassa - polacco" lang="pl" hreflang="pl" data-title="Twierdzenie Stone’a-Weierstrassa" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Stone-Weierstrass" title="Teorema de Stone-Weierstrass - portoghese" lang="pt" hreflang="pt" data-title="Teorema de Stone-Weierstrass" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%92%D0%B5%D0%B9%D0%B5%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D1%81%D0%B0_%E2%80%94_%D0%A1%D1%82%D0%BE%D1%83%D0%BD%D0%B0" title="Теорема Вейерштрасса — Стоуна - russo" lang="ru" hreflang="ru" data-title="Теорема Вейерштрасса — Стоуна" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_sats" title="Stone–Weierstrass sats - svedese" lang="sv" hreflang="sv" data-title="Stone–Weierstrass sats" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%92%D0%B5%D1%94%D1%80%D1%88%D1%82%D1%80%D0%B0%D1%81%D1%81%D0%B0_%E2%80%94_%D0%A1%D1%82%D0%BE%D1%83%D0%BD%D0%B0" title="Теорема Веєрштрасса — Стоуна - ucraino" lang="uk" hreflang="uk" data-title="Теорема Веєрштрасса — Стоуна" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%AD%8F%E5%B0%94%E6%96%BD%E7%89%B9%E6%8B%89%E6%96%AF%E9%80%BC%E8%BF%91%E5%AE%9A%E7%90%86" title="魏尔施特拉斯逼近定理 - cinese" lang="zh" hreflang="zh" data-title="魏尔施特拉斯逼近定理" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/Stone-Weierstrass%E5%AE%9A%E7%90%86" title="Stone-Weierstrass定理 - cantonese" lang="yue" hreflang="yue" data-title="Stone-Weierstrass定理" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q939927#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> 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href="https://www.wikidata.org/wiki/Special:EntityPage/Q939927" title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g"><span>Elemento Wikidata</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="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <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">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</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">Da Wikipedia, l'enciclopedia libera.</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="it" dir="ltr"><p>In <a href="/wiki/Analisi_matematica" title="Analisi matematica">analisi matematica</a>, il <b>teorema di approssimazione di Weierstrass</b> è un risultato che afferma che ogni <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzione</a> <a href="/wiki/Numero_reale" title="Numero reale">reale</a> <a href="/wiki/Funzione_continua" title="Funzione continua">continua</a> definita in un <a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">intervallo</a> <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">chiuso</a> e <a href="/wiki/Insieme_limitato" title="Insieme limitato">limitato</a> può essere <a href="/wiki/Approssimazione" title="Approssimazione">approssimata</a> a piacere con un <a href="/wiki/Polinomio" title="Polinomio">polinomio</a> di grado opportuno. </p><p>Questo è stato dimostrato da <a href="/wiki/Karl_Weierstra%C3%9F" class="mw-redirect" title="Karl Weierstraß">Karl Weierstraß</a> nel <a href="/wiki/1885" title="1885">1885</a>. Il teorema ha importanti risvolti sia teorici che pratici. <a href="/wiki/Marshall_Stone" title="Marshall Stone">Marshall Stone</a> lo ha generalizzato nel <a href="/wiki/1937" title="1937">1937</a>, allargando il <a href="/wiki/Dominio_(matematica)" class="mw-redirect" title="Dominio (matematica)">dominio</a> ad un certo tipo di <a href="/wiki/Spazio_topologico" title="Spazio topologico">spazio topologico</a> e non limitandosi ai polinomi come funzioni approssimanti. Il risultato generale è noto come <b>teorema di Stone-Weierstrass</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Enunciato">Enunciato</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=1" title="Modifica la sezione Enunciato" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=1" title="Edit section's source code: Enunciato"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data una <a href="/wiki/Funzione_continua" title="Funzione continua">funzione continua</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon [a,b]\longrightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon [a,b]\longrightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bda17a8ca7ea71906b37514dd0342131a206a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.641ex; height:2.843ex;" alt="{\displaystyle f\colon [a,b]\longrightarrow \mathbb {R} }"></span></dd></dl> <p>definita sull'intervallo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>, esiste una <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">successione</a> di polinomi </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\{p_{k}\right\}_{k\geq 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>{</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{p_{k}\right\}_{k\geq 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee3602b708d014c7050b4b68f0b977efa1b1346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.772ex; height:3.176ex;" alt="{\displaystyle \left\{p_{k}\right\}_{k\geq 1}}"></span></dd></dl> <p>tale che </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 \lim _{k\to +\infty }p_{k}(x)=f(x).}"> <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>k</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to +\infty }p_{k}(x)=f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/342da48cab237c26a4938da2cc34dc5cba0e6899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.368ex; height:4.176ex;" alt="{\displaystyle \lim _{k\to +\infty }p_{k}(x)=f(x).}"></span></dd></dl> <p>Il limite è da intendersi non solo puntualmente ma anche rispetto alla <a href="/wiki/Convergenza_uniforme" class="mw-redirect" title="Convergenza uniforme">convergenza uniforme</a> sul <a href="/wiki/Spazio_compatto" title="Spazio compatto">compatto</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=[a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=[a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7400ab11f38f0a5b3eeca7d44848c982bd5840d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.719ex; height:2.843ex;" alt="{\displaystyle K=[a,b]}"></span>, ossia con </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 \lim _{k\to +\infty }\sup _{x\in K}|f(x)-p_{k}(x)|=0.}"> <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>k</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>K</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to +\infty }\sup _{x\in K}|f(x)-p_{k}(x)|=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e8feb0a397ce5d8359b3f93ad95dd09df78970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.553ex; height:4.509ex;" alt="{\displaystyle \lim _{k\to +\infty }\sup _{x\in K}|f(x)-p_{k}(x)|=0.}"></span></dd></dl> <p>Una conseguenza immediata di questo teorema è che i polinomi sono <a href="/wiki/Insieme_denso" title="Insieme denso">densi</a> nello spazio delle funzioni continue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b823cba8d0220d7bd670bace88a3fe7fd07e6659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.642ex; height:2.843ex;" alt="{\displaystyle C(K)}"></span>, che quindi risulta essere uno <a href="/wiki/Spazio_separabile" title="Spazio separabile">spazio separabile</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Dimostrazione">Dimostrazione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=2" title="Modifica la sezione Dimostrazione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=2" title="Edit section's source code: Dimostrazione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Osservazioni_preliminari">Osservazioni preliminari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=3" title="Modifica la sezione Osservazioni preliminari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=3" title="Edit section's source code: Osservazioni preliminari"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Con la <a href="/wiki/Corrispondenza_biunivoca" title="Corrispondenza biunivoca">trasformazione biiettiva</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 {\begin{aligned}x&\mapsto {\frac {x-a}{b-a}}\\f(x)&\mapsto f(x)-f(a)-{\frac {x-a}{b-a}}(f(b)-f(a)),\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> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&\mapsto {\frac {x-a}{b-a}}\\f(x)&\mapsto f(x)-f(a)-{\frac {x-a}{b-a}}(f(b)-f(a)),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181e7b41738645ce1ac93c8ec4d45a1bf0113966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.547ex; margin-bottom: -0.291ex; width:43.135ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}x&\mapsto {\frac {x-a}{b-a}}\\f(x)&\mapsto f(x)-f(a)-{\frac {x-a}{b-a}}(f(b)-f(a)),\end{aligned}}}"></span></dd></dl> <p>il teorema può essere dimostrato, senza perdita di generalità, anche solo per funzioni che verificano la condizione </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]=[0,1],\quad f(0)=f(1)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]=[0,1],\quad f(0)=f(1)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db0756275e31281bd9c2e83bbc3d850e5966bee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.169ex; height:2.843ex;" alt="{\displaystyle [a,b]=[0,1],\quad f(0)=f(1)=0.}"></span></dd></dl> <p>Estendendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> ponendola uguale a zero al di fuori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> si ottiene una funzione <a href="/wiki/Continuit%C3%A0_uniforme" title="Continuità uniforme">uniformemente continua</a> su tutto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> (la funzione di partenza è uniformemente continua su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> per il <a href="/wiki/Teorema_di_Heine-Cantor" title="Teorema di Heine-Cantor">teorema di Heine-Cantor</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Definizione_e_proprietà_dei_polinomi"><span id="Definizione_e_propriet.C3.A0_dei_polinomi"></span>Definizione e proprietà dei polinomi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=4" title="Modifica la sezione Definizione e proprietà dei polinomi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=4" title="Edit section's source code: Definizione e proprietà dei polinomi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> <a href="/wiki/Numero_naturale" title="Numero naturale">numero naturale</a>, i polinomi </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 q_{k}\colon \mathbb {R} \rightarrow \mathbb {R} ,\qquad q_{k}(x)={\begin{cases}(k+1)(1-x)^{k}&x\in [0,1]\\0&x\in \mathbb {R} \setminus [0,1]\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mspace width="2em" /> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <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> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> <mtd> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{k}\colon \mathbb {R} \rightarrow \mathbb {R} ,\qquad q_{k}(x)={\begin{cases}(k+1)(1-x)^{k}&x\in [0,1]\\0&x\in \mathbb {R} \setminus [0,1]\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f21dac7caeb20318a9fffa16f046d0b8dab129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.939ex; height:6.176ex;" alt="{\displaystyle q_{k}\colon \mathbb {R} \rightarrow \mathbb {R} ,\qquad q_{k}(x)={\begin{cases}(k+1)(1-x)^{k}&x\in [0,1]\\0&x\in \mathbb {R} \setminus [0,1]\end{cases}}}"></span></dd></dl> <p>sono non negativi e <a href="/wiki/Funzione_monotona" title="Funzione monotona">monotoni</a> decrescenti in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8786b5ef9daedb24adb59e7825c4096d99a99648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.299ex; height:2.843ex;" alt="{\displaystyle [0,1].}"></span> La funzione <a href="/wiki/Integrale" title="Integrale">integrale</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 I_{k}(z)=\int _{0}^{z}q_{k}(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msubsup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{k}(z)=\int _{0}^{z}q_{k}(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b1677c5ee20a82281fe09fea816ae79ef78d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.771ex; height:5.843ex;" alt="{\displaystyle I_{k}(z)=\int _{0}^{z}q_{k}(x)dx}"></span></dd></dl> <p>è monotona crescente in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8786b5ef9daedb24adb59e7825c4096d99a99648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.299ex; height:2.843ex;" alt="{\displaystyle [0,1].}"></span> Vale la proprietà di normalizzazione: </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 \forall k:\int _{\mathbb {R} }q_{k}(x)dx=\int _{0}^{1}q_{k}(x)dx=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>k</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> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall k:\int _{\mathbb {R} }q_{k}(x)dx=\int _{0}^{1}q_{k}(x)dx=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebca21ddbe3ecbc947f004d50d822e4314869c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.072ex; height:6.176ex;" alt="{\displaystyle \forall k:\int _{\mathbb {R} }q_{k}(x)dx=\int _{0}^{1}q_{k}(x)dx=1.}"></span></dd></dl> <p>I polinomi che approssimano <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> sono le funzioni </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 p_{k}(x)=\int _{\mathbb {R} }f(x+t)q_{k}(t)dt=\int _{0}^{1}f(x+t)q_{k}(t)dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo 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>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <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>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{k}(x)=\int _{\mathbb {R} }f(x+t)q_{k}(t)dt=\int _{0}^{1}f(x+t)q_{k}(t)dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5016aa2aec001a3300ce5d61167e8ff9d763fb1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:49.19ex; height:6.176ex;" alt="{\displaystyle p_{k}(x)=\int _{\mathbb {R} }f(x+t)q_{k}(t)dt=\int _{0}^{1}f(x+t)q_{k}(t)dt.}"></span></dd></dl> <p>Si può dimostrare che si tratta effettivamente di polinomi operando il cambio di variabile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=t+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=t+x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124a1537a73506c885cf44e7f3ececbd9b47b294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.199ex; height:2.176ex;" alt="{\displaystyle s=t+x}"></span> all'interno del primo integrale ed utilizzando il <a href="/wiki/Teorema_binomiale" title="Teorema binomiale">teorema binomiale</a> nell'intervallo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> per calcolare i coefficienti. </p> <div class="mw-heading mw-heading3"><h3 id="Parte_principale">Parte principale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=5" title="Modifica la sezione Parte principale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=5" title="Edit section's source code: Parte principale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considerando la proprietà di normalizzazione e la <a href="/wiki/Integrale#Valore_assoluto" title="Integrale">disuguaglianza integrale</a> abbiamo che, per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>: </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 |p_{k}(x)-f(x)|=\left|\int _{0}^{1}f(x+t)q_{k}(t)dt-f(x)\cdot 1\right|=\left|\int _{0}^{1}f(x+t)q_{k}(t)dt-f(x)\int _{0}^{1}q_{k}(t)dt\right|\leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <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>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <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>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |p_{k}(x)-f(x)|=\left|\int _{0}^{1}f(x+t)q_{k}(t)dt-f(x)\cdot 1\right|=\left|\int _{0}^{1}f(x+t)q_{k}(t)dt-f(x)\int _{0}^{1}q_{k}(t)dt\right|\leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4540968c0ea82878b8a7408eae0e730c2241c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:91.333ex; height:6.509ex;" alt="{\displaystyle |p_{k}(x)-f(x)|=\left|\int _{0}^{1}f(x+t)q_{k}(t)dt-f(x)\cdot 1\right|=\left|\int _{0}^{1}f(x+t)q_{k}(t)dt-f(x)\int _{0}^{1}q_{k}(t)dt\right|\leq }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq \int _{0}^{1}|f(x+t)-f(x)|q_{k}(t)dt=\diamondsuit }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi mathvariant="normal">♢</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq \int _{0}^{1}|f(x+t)-f(x)|q_{k}(t)dt=\diamondsuit }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b471998ef0368d22070648bbd87413e5ebdd587f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.745ex; height:6.176ex;" alt="{\displaystyle \leq \int _{0}^{1}|f(x+t)-f(x)|q_{k}(t)dt=\diamondsuit }"></span></dd></dl> <p>Dalla definizione di continuità uniforme di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10535d1a7a971ffeeb216605cb846099fab2e653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}"></span> fissato <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 \varepsilon /2>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon /2>0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a7ff5d3534ae140095f83ddf838f7a3d51247a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle \varepsilon /2>0,}"></span> </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 \exists \ \delta \in (0,1):|t|<\delta \Rightarrow |f(x+t)-f(x)|<{\frac {\varepsilon }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mtext> </mtext> <mi>δ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \ \delta \in (0,1):|t|<\delta \Rightarrow |f(x+t)-f(x)|<{\frac {\varepsilon }{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d860b65290aadf6130379eb273d1274d35302e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.155ex; height:4.676ex;" alt="{\displaystyle \exists \ \delta \in (0,1):|t|<\delta \Rightarrow |f(x+t)-f(x)|<{\frac {\varepsilon }{2}}.}"></span></dd></dl> <p>In base al <a href="/wiki/Teorema_di_Weierstrass" title="Teorema di Weierstrass">teorema di Weierstrass</a> esiste il <a href="/wiki/Massimo_e_minimo_di_una_funzione" title="Massimo e minimo di una funzione">massimo</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 M=\max \left\{\left|f(x)\right|:x\in [0,1]\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\max \left\{\left|f(x)\right|:x\in [0,1]\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff32d9e43c42a838709cd20a83b94e1f10037fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.084ex; height:2.843ex;" alt="{\displaystyle M=\max \left\{\left|f(x)\right|:x\in [0,1]\right\}.}"></span></dd></dl> <p>Fatte queste considerazioni e tenendo presente la <a href="/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare">disuguaglianza triangolare</a>, la <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 \diamondsuit }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">♢</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \diamondsuit }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4b9f727c53c7a5146318355922aa861e1c6804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \diamondsuit }"></span> diventa: </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 \diamondsuit =\int _{0}^{\delta }|f(x+t)-f(x)|q_{k}(t)dt+\int _{\delta }^{1}|f(x+t)-f(x)|q_{k}(t)dt\leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">♢</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \diamondsuit =\int _{0}^{\delta }|f(x+t)-f(x)|q_{k}(t)dt+\int _{\delta }^{1}|f(x+t)-f(x)|q_{k}(t)dt\leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddb5ac182284c77ba52909f3129d591e1087ee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:64.89ex; height:6.343ex;" alt="{\displaystyle \diamondsuit =\int _{0}^{\delta }|f(x+t)-f(x)|q_{k}(t)dt+\int _{\delta }^{1}|f(x+t)-f(x)|q_{k}(t)dt\leq }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq {\frac {\varepsilon }{2}}\int _{0}^{\delta }q_{k}(t)dt+\int _{\delta }^{1}[|f(x+t)|+|f(x)|]q_{k}(t)dt\leq {\frac {\varepsilon }{2}}+2M\int _{\delta }^{1}q_{k}(t)dt\leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mn>2</mn> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msubsup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">]</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>M</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq {\frac {\varepsilon }{2}}\int _{0}^{\delta }q_{k}(t)dt+\int _{\delta }^{1}[|f(x+t)|+|f(x)|]q_{k}(t)dt\leq {\frac {\varepsilon }{2}}+2M\int _{\delta }^{1}q_{k}(t)dt\leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ca4ad58ff760954ec4acc9926c93089852c09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:73.038ex; height:6.343ex;" alt="{\displaystyle \leq {\frac {\varepsilon }{2}}\int _{0}^{\delta }q_{k}(t)dt+\int _{\delta }^{1}[|f(x+t)|+|f(x)|]q_{k}(t)dt\leq {\frac {\varepsilon }{2}}+2M\int _{\delta }^{1}q_{k}(t)dt\leq }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq {\frac {\varepsilon }{2}}+2M(k+1)\int _{\delta }^{1}(1-t)^{k}dt={\frac {\varepsilon }{2}}+2M(1-\delta )^{k+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>M</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>δ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq {\frac {\varepsilon }{2}}+2M(k+1)\int _{\delta }^{1}(1-t)^{k}dt={\frac {\varepsilon }{2}}+2M(1-\delta )^{k+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e6580be18d440ae3b1f9fc1fa9fd7bbcfd4330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.861ex; height:6.176ex;" alt="{\displaystyle \leq {\frac {\varepsilon }{2}}+2M(k+1)\int _{\delta }^{1}(1-t)^{k}dt={\frac {\varepsilon }{2}}+2M(1-\delta )^{k+1}.}"></span></dd></dl> <p>Dato che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\delta <1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>δ</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\delta <1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6905aed10a84e38bf594cd5e8b544ebe44612f67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.217ex; height:2.676ex;" alt="{\displaystyle 0<\delta <1,}"></span> il secondo termine nel secondo membro dell'ultima equazione tende a zero per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> che tende a infinito, perciò è minore di <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 \varepsilon /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f00c527a451f6daf695e866c39ff8d2a0544baa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.408ex; height:2.843ex;" alt="{\displaystyle \varepsilon /2}"></span> per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> sufficientemente grande. In definitiva: </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 \forall \varepsilon >0\ \exists k:|p_{k}(x)-f(x)|<\varepsilon ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mi>k</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\ \exists k:|p_{k}(x)-f(x)|<\varepsilon ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0eb5756a912edd34102f050a0df7fc0702fbc3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.436ex; height:2.843ex;" alt="{\displaystyle \forall \varepsilon >0\ \exists k:|p_{k}(x)-f(x)|<\varepsilon ,}"></span></dd></dl> <p>cioè </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 \lim _{k\to +\infty }p_{k}(x)=f(x).}"> <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>k</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to +\infty }p_{k}(x)=f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/342da48cab237c26a4938da2cc34dc5cba0e6899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.368ex; height:4.176ex;" alt="{\displaystyle \lim _{k\to +\infty }p_{k}(x)=f(x).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Caso_complesso">Caso complesso</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=6" title="Modifica la sezione Caso complesso" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=6" title="Edit section's source code: Caso complesso"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il teorema si può estendere a funzioni a valori <a href="/wiki/Numero_complesso" title="Numero complesso">complessi</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon [a,b]\rightarrow \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon [a,b]\rightarrow \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6a2fd3f560fa905dfeb326417ea5768565ea78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.16ex; height:2.843ex;" alt="{\displaystyle f\colon [a,b]\rightarrow \mathbb {C} }"></span></dd></dl> <p>continue. La dimostrazione è analoga al caso reale, tenendo presente, però, che gli integrali non sono quelli ordinari ma <a href="/wiki/Integrale_di_linea" title="Integrale di linea">sui cammini</a> e che al posto del <a href="/wiki/Valore_assoluto" title="Valore assoluto">valore assoluto</a> nelle formule abbiamo la <a href="/wiki/Valore_assoluto#Funzione_modulo" title="Valore assoluto">funzione modulo</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Enunciato_del_teorema_tramite_i_concetti_degli_spazi_normati">Enunciato del teorema tramite i concetti degli spazi normati</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=7" title="Modifica la sezione Enunciato del teorema tramite i concetti degli spazi normati" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=7" title="Edit section's source code: Enunciato del teorema tramite i concetti degli spazi normati"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Usando la terminologia degli <a href="/wiki/Spazio_normato" title="Spazio normato">spazi normati</a>, il teorema afferma che, con la <a href="/wiki/Norma_uniforme" title="Norma uniforme">norma uniforme</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{\infty }=\max \left\{\,\left|f(x)\right|:x\in [a,b]\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }=\max \left\{\,\left|f(x)\right|:x\in [a,b]\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ec10b87e04c7f39acd68fb5e2bddb5fc6f1f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.41ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }=\max \left\{\,\left|f(x)\right|:x\in [a,b]\right\},}"></span></dd></dl> <p>lo <a href="/wiki/Spazio_funzionale" title="Spazio funzionale">spazio funzionale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}([a,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">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}([a,b])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b9025311574a9e157a2f0fd9213a0cf388ced7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.068ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}([a,b])}"></span> dei polinomi sull'intervallo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> è <a href="/wiki/Insieme_denso" title="Insieme denso">denso</a> nello spazio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}^{0}([a,b])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{0}([a,b])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7875b8d1c80f23c07c5418bec48fc5a64e1a5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.662ex; height:3.176ex;" alt="{\displaystyle {\mathcal {C}}^{0}([a,b])}"></span> delle funzioni continue su tale intervallo. </p><p>Nella dimostrazione proposta abbiamo che la disuguaglianza </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 |p_{k}(x)-f(x)|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |p_{k}(x)-f(x)|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6972d1749cbc5ea5e3619749d68d1cecaf5c302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.131ex; height:2.843ex;" alt="{\displaystyle |p_{k}(x)-f(x)|<\varepsilon }"></span></dd></dl> <p>vale per qualsiasi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> quindi in particolare vale per </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 \max \left\{\,\left|p_{k}(x)-f(x)\right|:x\in [0,1]\right\}=\|p_{k}-f\|_{\infty }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max \left\{\,\left|p_{k}(x)-f(x)\right|:x\in [0,1]\right\}=\|p_{k}-f\|_{\infty }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd366f4ef22a1adebeaf000a461e7f6d0f778cba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.456ex; height:2.843ex;" alt="{\displaystyle \max \left\{\,\left|p_{k}(x)-f(x)\right|:x\in [0,1]\right\}=\|p_{k}-f\|_{\infty }.}"></span></dd></dl> <p>Perciò </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 \lim _{k\to \infty }\|p_{k}-f\|_{\infty }=0.}"> <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>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }\|p_{k}-f\|_{\infty }=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa118b470af94059d2d8fa89fd976e5ab0e20d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.015ex; height:4.009ex;" alt="{\displaystyle \lim _{k\to \infty }\|p_{k}-f\|_{\infty }=0.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Conseguenze">Conseguenze</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=8" title="Modifica la sezione Conseguenze" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=8" title="Edit section's source code: Conseguenze"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Risvolti_teorici">Risvolti teorici</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=9" title="Modifica la sezione Risvolti teorici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=9" title="Edit section's source code: Risvolti teorici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una prima conseguenza è che lo spazio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}^{0}([a,b])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{0}([a,b])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7875b8d1c80f23c07c5418bec48fc5a64e1a5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.662ex; height:3.176ex;" alt="{\displaystyle {\mathcal {C}}^{0}([a,b])}"></span> è <a href="/wiki/Spazio_separabile" title="Spazio separabile">separabile</a> perché <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}([a,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">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}([a,b])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b9025311574a9e157a2f0fd9213a0cf388ced7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.068ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}([a,b])}"></span> stesso è separabile, dato che contiene l'<a href="/wiki/Insieme_denso" title="Insieme denso">insieme denso</a> e <a href="/wiki/Insieme_numerabile" title="Insieme numerabile">numerabile</a> dei polinomi a coefficienti <a href="/wiki/Numero_razionale" title="Numero razionale">razionali</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 M=\left\{\sum _{i=0}^{n}q_{i}x^{i}:q_{i}\in \mathbb {Q} ,n\in \mathbb {N} \right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>:</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\left\{\sum _{i=0}^{n}q_{i}x^{i}:q_{i}\in \mathbb {Q} ,n\in \mathbb {N} \right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcc587dd83be4cf652aec7986659b0b8f791627" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.399ex; height:7.509ex;" alt="{\displaystyle M=\left\{\sum _{i=0}^{n}q_{i}x^{i}:q_{i}\in \mathbb {Q} ,n\in \mathbb {N} \right\}.}"></span></dd></dl> <p>Un'altra conseguenza è che è separabile qualsiasi insieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> in cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}^{0}([a,b])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{0}([a,b])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7875b8d1c80f23c07c5418bec48fc5a64e1a5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.662ex; height:3.176ex;" alt="{\displaystyle {\mathcal {C}}^{0}([a,b])}"></span> è denso. Tra i tanti esempi di insiemi che verificano questa condizione, si può citare lo <a href="/wiki/Spazio_Lp" title="Spazio Lp">spazio L<sup>1</sup></a> delle funzioni a modulo <a href="/wiki/Integrale_di_Lebesgue" title="Integrale di Lebesgue">integrabile secondo Lebesgue</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba5cb29655f824ce80a0b6a32d9326d0e8742cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.202ex; height:2.843ex;" alt="{\displaystyle [a,b].}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Risvolti_pratici">Risvolti pratici</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=10" title="Modifica la sezione Risvolti pratici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=10" title="Edit section's source code: Risvolti pratici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nella maggior parte dei problemi pratici in cui bisogna valutare una funzione sconosciuta, si sa che la funzione in questione è continua (o lo si ipotizza). Il teorema ci assicura, quindi, che possiamo sempre in linea di principio trovare un polinomio che approssima la funzione incognita con un grado di precisione arbitrario. Ovviamente altra cosa è determinare esplicitamente un <a href="/wiki/Algoritmo" title="Algoritmo">algoritmo</a> per calcolare questo polinomio. </p> <div class="mw-heading mw-heading2"><h2 id="Il_teorema_di_Stone-Weierstrass">Il teorema di Stone-Weierstrass</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=11" title="Modifica la sezione Il teorema di Stone-Weierstrass" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=11" title="Edit section's source code: Il teorema di Stone-Weierstrass"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> uno <a href="/wiki/Spazio_topologico" title="Spazio topologico">spazio topologico</a> <a href="/wiki/Spazio_di_Hausdorff" title="Spazio di Hausdorff">di Hausdorff</a> <a href="/wiki/Spazio_compatto" title="Spazio compatto">compatto</a> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(K,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99b1315c1b45db7d2efb02e9b6a5afe217771fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle C(K,\mathbb {C} )}"></span> l'<a href="/wiki/Algebra_su_campo" title="Algebra su campo">algebra</a> delle funzioni continue a valori complessi ivi definite, con la <a href="/wiki/Spazio_normato#Struttura_topologica" title="Spazio normato">topologia generata</a> dalla <a href="/wiki/Norma_uniforme" title="Norma uniforme">norma uniforme</a>. Questa è una <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a> dove lo *-operatore è rappresentato dal <a href="/wiki/Complesso_coniugato" title="Complesso coniugato">coniugio dei numeri complessi</a>. </p><p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq C(K,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq C(K,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1205ce469324806fbc1bf6e5586feb525ea7b525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.216ex; height:2.843ex;" alt="{\displaystyle B\subseteq C(K,\mathbb {C} )}"></span>. Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> è una sottoalgebra <a href="/wiki/Involuzione_(teoria_degli_insiemi)" title="Involuzione (teoria degli insiemi)">involutiva</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(K,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99b1315c1b45db7d2efb02e9b6a5afe217771fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle C(K,\mathbb {C} )}"></span> (cioè se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> è un <a href="/wiki/Sottospazio" class="mw-disambig" title="Sottospazio">sottospazio</a> <a href="/wiki/Chiusura" class="mw-disambig" title="Chiusura">chiuso</a> rispetto al prodotto e al coniugio in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>) che separa i punti di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>, cioè se vale la condizione </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 \forall x,y\in K\ \exists f\in B:x\neq y\Rightarrow f(x)\neq f(y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>K</mi> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mi>f</mi> <mo>∈</mo> <mi>B</mi> <mo>:</mo> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo stretchy="false">⇒</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in K\ \exists f\in B:x\neq y\Rightarrow f(x)\neq f(y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fad2f7ea851210570b8817b0c317fa86a385ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.016ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in K\ \exists f\in B:x\neq y\Rightarrow f(x)\neq f(y),}"></span></dd></dl> <p>allora la <a href="/w/index.php?title=*-algebra&action=edit&redlink=1" class="new" title="*-algebra (la pagina non esiste)">*-algebra</a> generata dall'<a href="/wiki/Elemento_neutro" title="Elemento neutro">unità</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> è <a href="/wiki/Insieme_denso" title="Insieme denso">densa</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(K,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99b1315c1b45db7d2efb02e9b6a5afe217771fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle C(K,\mathbb {C} )}"></span>. </p><p>La *-algebra in questione è un insieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> che contiene la funzione costante <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> e che, se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/888dd9afe888206b1b0a5cd2df0b7597bf5ada1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:2.509ex;" alt="{\displaystyle f\in X}"></span>, contiene qualsiasi altra funzione ottenuta partendo da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> e applicando un numero finito di volte le operazioni di addizione, moltiplicazione, coniugazione complessa o moltiplicazione per un numero complesso. </p><p>Il caso reale del teorema (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq C(K,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</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\subseteq C(K,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8013aec647d8b865867ab7f0bd9778e14bd72d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.216ex; height:2.843ex;" alt="{\displaystyle B\subseteq C(K,\mathbb {R} )}"></span>) si ottiene come caso particolare di quello complesso, perché se una successione di funzioni complesse <a href="/wiki/Convergenza_uniforme" class="mw-redirect" title="Convergenza uniforme">converge uniformemente</a> ad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> allora la successione delle <a href="/wiki/Parte_reale" title="Parte reale">parti reali</a> delle stesse funzioni converge uniformemente alla parte reale di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Ulteriori_generalizzazioni">Ulteriori generalizzazioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=12" title="Modifica la sezione Ulteriori generalizzazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=12" title="Edit section's source code: Ulteriori generalizzazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Esistono due ulteriori generalizzazioni del teorema. </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_di_Stone-Weierstrass_per_i_reticoli_di_funzioni_continue">Teorema di Stone-Weierstrass per i reticoli di funzioni continue</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=13" title="Modifica la sezione Teorema di Stone-Weierstrass per i reticoli di funzioni continue" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=13" title="Edit section's source code: Teorema di Stone-Weierstrass per i reticoli di funzioni continue"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La prima è la versione per <a href="/wiki/Reticolo_(matematica)" title="Reticolo (matematica)">reticoli</a> del teorema di Stone-Weierstass. </p><p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> uno spazio topologico di Hausdorff compatto costituito da almeno due punti e sia <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> un reticolo contenuto in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</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 C(K,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205bb9d13bee7d0bcc7569c8da9b6544dfedb913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle C(K,\mathbb {R} )}"></span> che verifica la condizione </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 \forall x,y\in K\ \forall a,b\in \mathbb {R} \ \exists f\in R:f(x)=a\wedge f(y)=b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>K</mi> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mtext> </mtext> <mi mathvariant="normal">∃</mi> <mi>f</mi> <mo>∈</mo> <mi>R</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>∧</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in K\ \forall a,b\in \mathbb {R} \ \exists f\in R:f(x)=a\wedge f(y)=b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff5478d10d25e262d59bc40f2c490aa4e9d8eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.38ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in K\ \forall a,b\in \mathbb {R} \ \exists f\in R:f(x)=a\wedge f(y)=b.}"></span></dd></dl> <p>Allora <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> è denso in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</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 C(K,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205bb9d13bee7d0bcc7569c8da9b6544dfedb913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle C(K,\mathbb {R} )}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Teorema_di_Bishop">Teorema di Bishop</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=14" title="Modifica la sezione Teorema di Bishop" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=14" title="Edit section's source code: Teorema di Bishop"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La seconda è un teorema dovuto a Errett Bishop. </p><p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> uno spazio topologico di Hausdorff compatto, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> una sottoalgebra chiusa dello <a href="/wiki/Spazio_di_Banach" title="Spazio di Banach">spazio di Banach</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(K,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99b1315c1b45db7d2efb02e9b6a5afe217771fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle C(K,\mathbb {C} )}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> una funzione appartenente a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(K,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(K,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99b1315c1b45db7d2efb02e9b6a5afe217771fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle C(K,\mathbb {C} )}"></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{/W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{/W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae9e43c9ef67887f6e11964a8ec686c335c110f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.915ex; height:3.009ex;" alt="{\displaystyle f_{/W}}"></span> indica una <a href="/wiki/Restrizione_di_una_funzione" title="Restrizione di una funzione">restrizione</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> su un sottoinsieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\subseteq K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>⊆</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\subseteq K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54307094e38eb85f4a603d3b8c0e20d217f2c464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.6ex; height:2.343ex;" alt="{\displaystyle W\subseteq K}"></span>, mentre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{/W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{/W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d9632a743a9c197b7d351960b470c0744ad062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.519ex; height:3.009ex;" alt="{\displaystyle A_{/W}}"></span> indica lo spazio delle restrizioni su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> di funzioni appartenenti ad <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>.<br /> Sia <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 R\subset C(K,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⊂</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\subset C(K,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129c6b46745b69aef9c83abbfee3faf6a8bc51f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.216ex; height:2.843ex;" alt="{\displaystyle R\subset C(K,\mathbb {C} )}"></span> il sottoinsieme delle <a href="/wiki/Funzione_costante" title="Funzione costante">funzioni costanti</a> reali. Consideriamo l'insieme </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\left\{X\subset K:A_{/X}\subseteq R_{/X}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>X</mi> <mo>⊂</mo> <mi>K</mi> <mo>:</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>X</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>X</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\left\{X\subset K:A_{/X}\subseteq R_{/X}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67de910ee31cc18538549db031a0f1e3232298d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.192ex; height:3.343ex;" alt="{\displaystyle V=\left\{X\subset K:A_{/X}\subseteq R_{/X}\right\}}"></span></dd></dl> <p>e chiamiamo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4a4528b6277b1b85d6a42e87a502f472902f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.314ex; height:2.509ex;" alt="{\displaystyle V_{M}}"></span> il sottoinsieme degli insiemi massimali di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> secondo l'<a href="/wiki/Inclusione_(matematica)" title="Inclusione (matematica)">inclusione insiemistica</a>. Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> verifica la condizione </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 \forall S\in V_{M}:f_{/S}\in A_{/S},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>S</mi> <mo>∈</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall S\in V_{M}:f_{/S}\in A_{/S},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8b4a9893209b2ad0aa2c69787d019a6a84d365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.483ex; height:3.009ex;" alt="{\displaystyle \forall S\in V_{M}:f_{/S}\in A_{/S},}"></span></dd></dl> <p>allora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65aa8d2f370727a799a1c413e553afad12518189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.862ex; height:2.509ex;" alt="{\displaystyle f\in A}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=15" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=15" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) Charles Jean de la Vallée Poussin <i><a rel="nofollow" class="external text" href="http://name.umdl.umich.edu/ACQ7767.0001.001">Leçons sur l'approximation des fonctions d'une variable réelle</a></i> (Parigi, Gauthier-Villars, 1919)</li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Bertran Walsh <i><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080828160958/http://www.math.rutgers.edu/courses/501/501-f99/stnw_501.pdf">The Stone-Weierstrass theorem</a></i> (Rutgers University, New Jersey, USA, 1999)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=16" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=16" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebra_su_campo" title="Algebra su campo">Algebra su campo</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Continuit%C3%A0_uniforme" title="Continuità uniforme">Continuità uniforme</a></li> <li><a href="/wiki/Funzione_continua" title="Funzione continua">Funzione continua</a></li> <li><a href="/wiki/Integrale" title="Integrale">Integrale</a></li> <li><a href="/wiki/Interpolazione_polinomiale" title="Interpolazione polinomiale">Interpolazione polinomiale</a></li> <li><a href="/wiki/Polinomio" title="Polinomio">Polinomio</a></li> <li><a href="/wiki/Reticolo_(matematica)" title="Reticolo (matematica)">Reticolo (matematica)</a></li> <li><a href="/wiki/Spazio_normato" title="Spazio normato">Spazio normato</a></li> <li><a href="/wiki/Spazio_separabile" title="Spazio separabile">Spazio separabile</a></li> <li><a href="/wiki/Spazio_topologico" title="Spazio topologico">Spazio topologico</a></li> <li><a href="/wiki/Teorema_di_Heine-Cantor" title="Teorema di Heine-Cantor">Teorema di Heine-Cantor</a></li> <li><a href="/wiki/Teorema_di_Weierstrass" title="Teorema di Weierstrass">Teorema di Weierstrass</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&veaction=edit&section=17" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teorema_di_approssimazione_di_Weierstrass&action=edit&section=17" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/teorema-di-stone-weierstrass_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">Stone-Weierstrass, teorema di</span></a>, in <span style="font-style:italic;">Enciclopedia della Matematica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2013.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q939927#P9621" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Stone-WeierstrassTheorem.html"><span style="font-style:italic;">Teorema di approssimazione di Weierstrass</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q939927#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Analisi_matematica"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Analisi_matematica" title="Template:Analisi matematica"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Analisi_matematica&action=edit&redlink=1" class="new" title="Discussioni template:Analisi matematica (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Analisi_matematica&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Analisi_matematica" title="Analisi matematica">Analisi matematica</a></span></th></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Calcolo_infinitesimale" title="Calcolo infinitesimale">Calcolo infinitesimale</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Numero_reale" title="Numero reale">Numero reale</a><b> ·</b> <a href="/wiki/Infinitesimo" title="Infinitesimo">Infinitesimo</a><b> ·</b> <a href="/wiki/O-grande" title="O-grande">O-grande</a><b> ·</b> <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">Successione</a> (<a href="/wiki/Successione_di_funzioni" title="Successione di funzioni">di funzioni</a>)<b> ·</b> <a href="/wiki/Successione_di_Cauchy" title="Successione di Cauchy">Successione di Cauchy</a><b> ·</b> <a href="/wiki/Teorema_di_Bolzano-Weierstrass" title="Teorema di Bolzano-Weierstrass">Teorema di Bolzano-Weierstrass</a><b> ·</b> <a href="/wiki/Stima_asintotica" title="Stima asintotica">Stima asintotica</a><b> ·</b> <a href="/wiki/Limite_(matematica)" title="Limite (matematica)">Limite</a> (<a href="/wiki/Limite_di_una_funzione" title="Limite di una funzione">di una funzione</a><b> ·</b> <a href="/wiki/Limite_di_una_successione" title="Limite di una successione">di una successione</a><b> ·</b> <a href="/wiki/Forma_indeterminata" title="Forma indeterminata">Forma indeterminata</a>)<b> ·</b> <a href="/wiki/Teorema_del_confronto" title="Teorema del confronto">Teorema dei due carabinieri</a><b> ·</b> <a href="/wiki/Limite_notevole" title="Limite notevole">Limite notevole</a><b> ·</b> <a href="/wiki/Punto_di_accumulazione" title="Punto di accumulazione">Punto di accumulazione</a><b> ·</b> <a href="/wiki/Punto_isolato" title="Punto isolato">Punto isolato</a><b> ·</b> <a href="/wiki/Intorno" title="Intorno">Intorno</a><b> ·</b> <a href="/wiki/Serie_(matematica)" title="Serie (matematica)">Serie</a> (<a href="/wiki/Serie_di_funzioni" title="Serie di funzioni">di funzioni</a>)<b> ·</b> <a href="/wiki/Criteri_di_convergenza" title="Criteri di convergenza">Criteri di convergenza</a><b> ·</b> <a href="/wiki/Limite_di_funzioni_a_pi%C3%B9_variabili" title="Limite di funzioni a più variabili">Limite di funzioni a più variabili</a></td><td rowspan="7" class="navbox_image"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_kmplot.svg" class="mw-file-description" title="Analisi matematica"><img alt="Analisi matematica" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/100px-Nuvola_apps_kmplot.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/150px-Nuvola_apps_kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/200px-Nuvola_apps_kmplot.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>Analisi matematica</figcaption></figure></td></tr><tr><th colspan="1" class="navbox_group">Studio della continuità</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Funzione_continua" title="Funzione continua">Funzione continua</a><b> ·</b> <a href="/wiki/Punto_di_discontinuit%C3%A0" title="Punto di discontinuità">Punto di discontinuità</a><b> ·</b> <a href="/wiki/Continuit%C3%A0_uniforme" title="Continuità uniforme">Continuità uniforme</a><b> ·</b> <a href="/wiki/Funzione_lipschitziana" title="Funzione lipschitziana">Funzione lipschitziana</a><b> ·</b> <a href="/wiki/Teorema_di_Bolzano" title="Teorema di Bolzano">Teorema di Bolzano</a><b> ·</b> <a href="/wiki/Teorema_di_Weierstrass" title="Teorema di Weierstrass">Teorema di Weierstrass</a><b> ·</b> <a href="/wiki/Teorema_dei_valori_intermedi" title="Teorema dei valori intermedi">Teorema dei valori intermedi</a><b> ·</b> <a href="/wiki/Teorema_di_Heine-Cantor" title="Teorema di Heine-Cantor">Teorema di Heine-Cantor</a><b> ·</b> <a href="/wiki/Modulo_di_continuit%C3%A0" title="Modulo di continuità">Modulo di continuità</a><b> ·</b> <a href="/wiki/Funzione_semicontinua" class="mw-redirect" title="Funzione semicontinua">Funzione semicontinua</a><b> ·</b> <a href="/wiki/Continuit%C3%A0_separata" title="Continuità separata">Continuità separata</a><b> ·</b> <a class="mw-selflink selflink">Teorema di approssimazione di Weierstrass</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Calcolo_differenziale" class="mw-redirect" title="Calcolo differenziale">Calcolo differenziale</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Derivata" title="Derivata">Derivata</a><b> ·</b> <a href="/wiki/Differenziale_(matematica)" title="Differenziale (matematica)">Differenziale</a><b> ·</b> <a href="/wiki/Regole_di_derivazione" title="Regole di derivazione">Regole di derivazione</a><b> ·</b> <a href="/wiki/Teorema_di_Fermat_sui_punti_stazionari" title="Teorema di Fermat sui punti stazionari">Teorema di Fermat</a><b> ·</b> <a href="/wiki/Teorema_di_Rolle" title="Teorema di Rolle">Teorema di Rolle</a><b> ·</b> <a href="/wiki/Teorema_di_Lagrange" title="Teorema di Lagrange">Teorema di Lagrange</a><b> ·</b> <a href="/wiki/Teorema_di_Cauchy_(analisi_matematica)" title="Teorema di Cauchy (analisi matematica)">Teorema di Cauchy</a><b> ·</b> <a href="/wiki/Teorema_di_Darboux" title="Teorema di Darboux">Teorema di Darboux</a><b> ·</b> <a href="/wiki/Teorema_di_Taylor" title="Teorema di Taylor">Teorema di Taylor</a><b> ·</b> <a href="/wiki/Serie_di_Taylor" title="Serie di Taylor">Serie di Taylor</a><b> ·</b> <a href="/wiki/Funzione_differenziabile" title="Funzione differenziabile">Funzione differenziabile</a><b> ·</b> <a href="/wiki/Gradiente_(funzione)" class="mw-redirect" title="Gradiente (funzione)">Gradiente</a><b> ·</b> <a href="/wiki/Matrice_jacobiana" title="Matrice jacobiana">Jacobiana</a><b> ·</b> <a href="/wiki/Matrice_hessiana" title="Matrice hessiana">Hessiana</a><b> ·</b> <a href="/wiki/Forma_differenziale" title="Forma differenziale">Forma differenziale</a><b> ·</b> <a href="/wiki/Generalizzazioni_della_derivata" title="Generalizzazioni della derivata">Generalizzazioni della derivata</a><b> ·</b> <a href="/wiki/Derivata_parziale" title="Derivata parziale">Derivata parziale</a><b> ·</b> <a href="/wiki/Derivata_mista" title="Derivata mista">Derivata mista</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Integrale" title="Integrale">Integrale</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Primitiva_(matematica)" title="Primitiva (matematica)">Primitiva</a><b> ·</b> <a href="/wiki/Integrale_di_Riemann" title="Integrale di Riemann">Integrale di Riemann</a><b> ·</b> <a href="/wiki/Integrale_improprio" title="Integrale improprio">Integrale improprio</a><b> ·</b> <a href="/wiki/Integrale_di_Lebesgue" title="Integrale di Lebesgue">Integrale di Lebesgue</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_del_calcolo_integrale" title="Teorema fondamentale del calcolo integrale">Teorema fondamentale</a><b> ·</b> <a href="/wiki/Metodi_di_integrazione" title="Metodi di integrazione">Metodi di integrazione</a><b> ·</b> <a href="/wiki/Categoria:Tavole_di_integrali" title="Categoria:Tavole di integrali">Tavole</a><b> ·</b> <a href="/wiki/Integrale_multiplo" title="Integrale multiplo">Integrale multiplo</a>, <a href="/wiki/Integrale_di_linea" title="Integrale di linea">di linea</a> (<a href="/wiki/Integrale_di_linea_di_prima_specie" title="Integrale di linea di prima specie">1ª specie</a><b> ·</b> <a href="/wiki/Integrale_di_linea_di_seconda_specie" title="Integrale di linea di seconda specie">2ª specie</a>) e <a href="/wiki/Integrale_di_superficie" title="Integrale di superficie">di superficie</a> (<a href="/wiki/Integrale_di_volume" title="Integrale di volume">di volume</a>)</td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Studio_di_funzione" title="Studio di funzione">Studio di funzione</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">Funzione</a><b> ·</b> <a href="/wiki/Variabile_(matematica)" title="Variabile (matematica)">Variabile</a><b> ·</b> <a href="/wiki/Dominio_e_codominio" title="Dominio e codominio">Dominio e codominio</a><b> ·</b> <a href="/wiki/Funzioni_pari_e_dispari" title="Funzioni pari e dispari">Funzioni pari e dispari</a><b> ·</b> <a href="/wiki/Funzione_periodica" title="Funzione periodica">Funzione periodica</a><b> ·</b> <a href="/wiki/Funzione_monotona" title="Funzione monotona">Funzione monotona</a><b> ·</b> <a href="/wiki/Funzione_convessa" title="Funzione convessa">Funzione convessa</a><b> ·</b> <a href="/wiki/Massimo_e_minimo_di_una_funzione" title="Massimo e minimo di una funzione">Massimo e minimo di una funzione</a><b> ·</b> <a href="/wiki/Punto_angoloso" title="Punto angoloso">Punto angoloso</a><b> ·</b> <a href="/wiki/Cuspide_(matematica)" title="Cuspide (matematica)">Cuspide</a><b> ·</b> <a href="/wiki/Punto_di_flesso" title="Punto di flesso">Punto di flesso</a><b> ·</b> <a href="/wiki/Asintoto" title="Asintoto">Asintoto</a><b> ·</b> <a href="/wiki/Grafico_di_una_funzione" title="Grafico di una funzione">Grafico di una funzione</a><b> ·</b> <a href="/wiki/Funzione_iniettiva" title="Funzione iniettiva">Funzione iniettiva</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Disuguaglianza" title="Disuguaglianza">Disuguaglianze</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare">Disuguaglianza triangolare</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Cauchy-Schwarz" title="Disuguaglianza di Cauchy-Schwarz">Disuguaglianza di Cauchy-Schwarz</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Bernoulli" title="Disuguaglianza di Bernoulli">Bernoulli</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Jensen" title="Disuguaglianza di Jensen">Jensen</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_H%C3%B6lder" title="Disuguaglianza di Hölder">Hölder</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Young" title="Disuguaglianza di Young">Young</a><b> ·</b> <a href="/wiki/Disuguaglianza_di_Minkowski" title="Disuguaglianza di Minkowski">Minkowski</a></td></tr><tr><th colspan="1" class="navbox_group">Altro</th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Approssimazione_di_Stirling" title="Approssimazione di Stirling">Approssimazione di Stirling</a><b> ·</b> <a href="/wiki/Prodotto_di_Wallis" title="Prodotto di Wallis">Prodotto di Wallis</a><b> ·</b> <a href="/wiki/Funzione_Gamma" title="Funzione Gamma">Funzione Gamma</a><b> ·</b> <a href="/wiki/Teorema_delle_funzioni_implicite" title="Teorema delle funzioni implicite">Teorema delle funzioni implicite</a><b> ·</b> <a href="/wiki/Teorema_della_funzione_inversa" title="Teorema della funzione inversa">Teorema della funzione inversa</a><b> ·</b> <a href="/wiki/Condizione_di_H%C3%B6lder" title="Condizione di Hölder">Funzione hölderiana</a><b> ·</b> <a href="/wiki/Spazio_metrico" title="Spazio metrico">Spazio metrico</a><b> ·</b> <a href="/wiki/Spazio_normato" title="Spazio normato">Spazio normato</a><b> ·</b> <a href="/wiki/Intervallo_(matematica)" title="Intervallo (matematica)">Intervallo</a><b> ·</b> <a href="/wiki/Insieme_trascurabile" title="Insieme trascurabile">Insieme trascurabile</a><b> ·</b> <a href="/wiki/Insieme_chiuso" title="Insieme chiuso">Insieme chiuso</a><b> ·</b> <a href="/wiki/Insieme_aperto" title="Insieme aperto">Insieme aperto</a><b> ·</b> <a href="/wiki/Palla_(matematica)" title="Palla (matematica)">Palla</a><b> ·</b> <a href="/wiki/Omeomorfismo" title="Omeomorfismo">Omeomorfismo</a><b> ·</b> <a href="/wiki/Omeomorfismo_locale" title="Omeomorfismo locale">Omeomorfismo locale</a><b> ·</b> <a href="/wiki/Diffeomorfismo" title="Diffeomorfismo">Diffeomorfismo</a><b> ·</b> <a href="/wiki/Diffeomorfismo_locale" title="Diffeomorfismo locale">Diffeomorfismo locale</a><b> ·</b> <a href="/wiki/Classe_C_di_una_funzione" title="Classe C di una funzione">Classe C di una funzione</a><b> ·</b> <a href="/wiki/Equazione_differenziale" title="Equazione differenziale">Equazione differenziale</a><b> ·</b> <a href="/wiki/Problema_di_Cauchy" title="Problema di Cauchy">Problema di Cauchy</a></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output 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style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4341745-0">4341745-0</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb15014506g">cb15014506g</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb15014506g">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007546739505171">987007546739505171</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output 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Teorema di approssimazione di Weierstrass - Wikipedia