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Beschikbaar in 22 talen" > <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-22" 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">22 talen</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_%D9%87%D8%A7%D9%86-%D8%A8%D8%A7%D9%86%D8%A7%D8%AE" title="مبرهنة هان-باناخ – Arabisch" lang="ar" hreflang="ar" data-title="مبرهنة هان-باناخ" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Xan-Banax_teoremi" title="Xan-Banax teoremi – Azerbeidzjaans" lang="az" hreflang="az" data-title="Xan-Banax teoremi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbeidzjaans" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Hahn-Banach" title="Teorema de Hahn-Banach – Catalaans" lang="ca" hreflang="ca" data-title="Teorema de Hahn-Banach" data-language-autonym="Català" data-language-local-name="Catalaans" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hahnova%E2%80%93Banachova_v%C4%9Bta" title="Hahnova–Banachova věta – Tsjechisch" lang="cs" hreflang="cs" data-title="Hahnova–Banachova věta" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Hahn-Banach" title="Satz von Hahn-Banach – Duits" lang="de" hreflang="de" data-title="Satz von Hahn-Banach" data-language-autonym="Deutsch" data-language-local-name="Duits" 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/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem – Engels" lang="en" hreflang="en" data-title="Hahn–Banach theorem" data-language-autonym="English" data-language-local-name="Engels" 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_Hahn%E2%80%93Banach" title="Teorema de Hahn–Banach – Spaans" lang="es" hreflang="es" data-title="Teorema de Hahn–Banach" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D9%87%D8%A7%D9%86%E2%80%93%D8%A8%D8%A7%D9%86%D8%A7%D8%AE" title="قضیه هان–باناخ – Perzisch" lang="fa" hreflang="fa" data-title="قضیه هان–باناخ" data-language-autonym="فارسی" data-language-local-name="Perzisch" class="interlanguage-link-target"><span>فارسی</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_Hahn-Banach" title="Théorème de Hahn-Banach – Frans" lang="fr" hreflang="fr" data-title="Théorème de Hahn-Banach" data-language-autonym="Français" data-language-local-name="Frans" 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%90%D7%9F-%D7%91%D7%A0%D7%9A" title="משפט האן-בנך – Hebreeuws" lang="he" hreflang="he" data-title="משפט האן-בנך" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Hahn-Banach" title="Teorema di Hahn-Banach – Italiaans" lang="it" hreflang="it" data-title="Teorema di Hahn-Banach" data-language-autonym="Italiano" data-language-local-name="Italiaans" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8F%E3%83%BC%E3%83%B3%E2%80%93%E3%83%90%E3%83%8A%E3%83%83%E3%83%8F%E3%81%AE%E5%AE%9A%E7%90%86" title="ハーン–バナッハの定理 – Japans" lang="ja" hreflang="ja" data-title="ハーン–バナッハの定理" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%9C-%EB%B0%94%EB%82%98%ED%9D%90_%EC%A0%95%EB%A6%AC" title="한-바나흐 정리 – Koreaans" lang="ko" hreflang="ko" data-title="한-바나흐 정리" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Hahna-Banacha" title="Twierdzenie Hahna-Banacha – Pools" lang="pl" hreflang="pl" data-title="Twierdzenie Hahna-Banacha" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_%C3%ABd_Hahn-Banach" title="Teorema ëd Hahn-Banach – Piëmontees" lang="pms" hreflang="pms" data-title="Teorema ëd Hahn-Banach" data-language-autonym="Piemontèis" data-language-local-name="Piëmontees" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Hahn-Banach" title="Teorema de Hahn-Banach – Portugees" lang="pt" hreflang="pt" data-title="Teorema de Hahn-Banach" data-language-autonym="Português" data-language-local-name="Portugees" 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%A5%D0%B0%D0%BD%D0%B0_%E2%80%94_%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%B0" title="Теорема Хана — Банаха – Russisch" lang="ru" hreflang="ru" data-title="Теорема Хана — Банаха" data-language-autonym="Русский" data-language-local-name="Russisch" 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/Hahn-Banachs_sats" title="Hahn-Banachs sats – Zweeds" lang="sv" hreflang="sv" data-title="Hahn-Banachs sats" data-language-autonym="Svenska" data-language-local-name="Zweeds" 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%93%D0%B0%D0%BD%D0%B0_%E2%80%94_%D0%91%D0%B0%D0%BD%D0%B0%D1%85%D0%B0" title="Теорема Гана — Банаха – Oekraïens" lang="uk" hreflang="uk" data-title="Теорема Гана — Банаха" data-language-autonym="Українська" data-language-local-name="Oekraïens" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Hahn-Banach" title="Định lý Hahn-Banach – Vietnamees" lang="vi" hreflang="vi" data-title="Định lý Hahn-Banach" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamees" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%93%88%E6%81%A9-%E5%B7%B4%E6%8B%BF%E8%B5%AB%E5%AE%9A%E7%90%86" title="哈恩-巴拿赫定理 – Chinees" lang="zh" hreflang="zh" data-title="哈恩-巴拿赫定理" data-language-autonym="中文" data-language-local-name="Chinees" 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/%E5%93%88%E6%81%A9-%E5%B7%B4%E6%8B%BF%E8%B5%AB%E5%AE%9A%E7%90%86" title="哈恩-巴拿赫定理 – Kantonees" lang="yue" hreflang="yue" data-title="哈恩-巴拿赫定理" data-language-autonym="粵語" data-language-local-name="Kantonees" 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 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Uiterlijk</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">naar zijbalk verplaatsen</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">verbergen</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">Uit Wikipedia, de vrije encyclopedie</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="nl" dir="ltr"><p>In de <a href="/wiki/Functionaalanalyse" title="Functionaalanalyse">functionaalanalyse</a>, een deelgebied van de <a href="/wiki/Wiskunde" title="Wiskunde">wiskunde</a>, is de <b>stelling van Hahn-Banach</b> een centrale stelling. De stelling van Hahn-Banach staat de uitbreiding van <a href="/w/index.php?title=Begrensde_lineaire_operator&action=edit&redlink=1" class="new" title="Begrensde lineaire operator (de pagina bestaat niet)">begrensde lineaire operatoren</a> toe. Deze begrensde lineaire operatoren worden gedefinieerd op een <a href="/wiki/Deelruimtetopologie" title="Deelruimtetopologie">deelruimte</a>, die bestaat uit een aantal <a href="/wiki/Vectorruimte" title="Vectorruimte">vectorruimten</a>, op de gehele <a href="/wiki/Ruimte_(wiskunde)" title="Ruimte (wiskunde)">ruimten</a>. De stelling laat ook zien dat er "genoeg" <a href="/wiki/Continue_functie_(topologie)" title="Continue functie (topologie)">continue</a> <a href="/wiki/Lineaire_functionaal" title="Lineaire functionaal">lineaire functionalen</a> zijn gedefinieerd op elke <a href="/wiki/Genormeerde_vectorruimte" class="mw-redirect" title="Genormeerde vectorruimte">genormeerde vectorruimte</a> om de studie van de <a href="/wiki/Duale_ruimte" title="Duale ruimte">duale vectorruimten</a> interessant te maken. Een andere versie van de stelling van Hahn-Banach staat bekend als de <i>scheidingsstelling van Hahn-Banach</i> (of de <i>scheiden van het hypervlak stelling</i>) en heeft tal van toepassingen in de <a href="/wiki/Convexe_meetkunde" title="Convexe meetkunde">convexe meetkunde</a>. De stelling is vernoemd naar <a href="/wiki/Hans_Hahn" title="Hans Hahn">Hans Hahn</a> en <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a>, die de stelling eind jaren twintig van de twintigste eeuw onafhankelijk van elkaar bewezen. Ironisch genoeg werd de stelling al eerder, in 1912, door <a href="/wiki/Eduard_Helly" title="Eduard Helly">Eduard Helly</a> bewezen.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Meetkundige_vorm">Meetkundige vorm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stelling_van_Hahn-Banach&veaction=edit&section=1" title="Bewerk dit kopje: Meetkundige vorm" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Stelling_van_Hahn-Banach&action=edit&section=1" title="De broncode bewerken van de sectie: Meetkundige vorm"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zij <span class="mwe-math-element"><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> een <a href="/wiki/Open_verzameling" title="Open verzameling">open verzameling</a> in een reële <a href="/wiki/Banachruimte" title="Banachruimte">Banachruimte</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 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> en zij <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> een punt van <span class="mwe-math-element"><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> dat niet tot <span class="mwe-math-element"><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> behoort; dan bestaat er een hypervlak met vergelijking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b402ee5eeecca23892f2a10a9bedf886324672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle g(x)=c}"></span> waarvoor <span class="mwe-math-element"><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> aan de ene kant ligt (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(a)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(a)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099407399c3fb6c1d8ed33fd3e479434cfe552f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.416ex; height:2.843ex;" alt="{\displaystyle g(a)>0}"></span> voor alle <span class="mwe-math-element"><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\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span>) en <span class="mwe-math-element"><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> aan de andere kant (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ad56a392dee775a26c5a7eaf570310d2dcc0be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.516ex; height:2.843ex;" alt="{\displaystyle g(x)\leq 0}"></span>).<sup id="cite_ref-dieudonne_2-0" class="reference"><a href="#cite_note-dieudonne-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="toccolours appendix" role="presentation" style="font-size:90%; margin:1em 0 -0.5em; clear:both;"> <div><span style="font-weight:bold">Bronnen, noten en/of referenties</span></div> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Helly.html">http://www-history.mcs.st-andrews.ac.uk/Biographies/Helly.html</a> Helly op <a href="/wiki/MacTutor" class="mw-redirect" title="MacTutor">MacTutor</a></span> </li> <li id="cite_note-dieudonne-2"><span class="mw-cite-backlink"><a href="#cite_ref-dieudonne_2-0">↑</a></span> <span class="reference-text">Hoofdstuk 12, oefening 4c in Dieudonné, Jean, "Eléments d'analyse" deel 2, éditions Jacques Gabay, herdruk in 2005 van de 3de uitgave uit 1983.</span> </li> </ol></div></div> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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