@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-S7FM9BJ5-N
  skos:prefLabel "Hilbert space"@en, "espace de Hilbert"@fr ;
  a skos:Concept ;
  skos:related psr:-W4X7D58C-P .

psr:-HX2VX066-P
  skos:prefLabel "functional analysis"@en, "analyse fonctionnelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-W4X7D58C-P .

psr:-ZTD7VMDS-3
  skos:prefLabel "analyse convexe"@fr, "convex analysis"@en ;
  a skos:Concept ;
  skos:narrower psr:-W4X7D58C-P .

psr: a skos:ConceptScheme .
psr:-W4X7D58C-P
  skos:prefLabel "théorème de projection orthogonale sur un convexe fermé"@fr, "Hilbert projection theorem"@en ;
  skos:broader psr:-ZTD7VMDS-3, psr:-HX2VX066-P ;
  skos:definition """En mathématiques, le théorème de projection orthogonale sur un convexe fermé est un résultat de minimisation de la distance dont le principal corollaire est l'existence d'un supplémentaire orthogonal, donc d'une projection orthogonale sur un sous-espace vectoriel fermé. Dans le cadre particulier d'un espace de Hilbert, il remplace avantageusement le théorème de Hahn-Banach. Il est en effet plus simple à démontrer et plus puissant dans ses conséquences. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_projection_sur_un_convexe_ferm%C3%A9">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_projection_sur_un_convexe_ferm%C3%A9</a>)"""@fr, """In mathematics, the <b>Hilbert projection theorem</b> is a famous result of convex analysis that says that for every vector <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" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> in a Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle H}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>H</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle H}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="H"></span> and every nonempty closed convex <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\\\\subseteq H,}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>C</mi>
         <mo>⊆<!-- ⊆ --></mo>
         <mi>H</mi>
         <mo>,</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle C\\\\subseteq H,}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab52085a368cb16b96208ffec34c35698fceb12" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.575ex; height:2.509ex;" alt="{\\\\displaystyle C\\\\subseteq H,}"></span> there exists a unique vector <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\\\\in C}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>m</mi>
         <mo>∈<!-- ∈ --></mo>
         <mi>C</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle m\\\\in C}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7522269d6d29898498443db0d387a52041b10b44" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.647ex; height:2.176ex;" alt="{\\\\displaystyle m\\\\in C}"></span> for which <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-x\\\\|}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
         <mi>c</mi>
         <mo>−<!-- − --></mo>
         <mi>x</mi>
         <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\|c-x\\\\|}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f03903b52a66735073c12db1c4eeb1d018134d05" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.502ex; height:2.843ex;" alt="{\\\\displaystyle \\\\|c-x\\\\|}"></span> is minimized over the vectors <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\\\\in C}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>c</mi>
         <mo>∈<!-- ∈ --></mo>
         <mi>C</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle c\\\\in C}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447d3982c94c23d6b6d01c90da81a6125aa26567" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.614ex; height:2.176ex;" alt="c\\\\in C"></span>; that is, such that <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-x\\\\|\\\\leq \\\\|c-x\\\\|}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
         <mi>m</mi>
         <mo>−<!-- − --></mo>
         <mi>x</mi>
         <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
         <mo>≤<!-- ≤ --></mo>
         <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
         <mi>c</mi>
         <mo>−<!-- − --></mo>
         <mi>x</mi>
         <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\|m-x\\\\|\\\\leq \\\\|c-x\\\\|}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed71e6009126dd5bf1f7fa2f36365928c7ea269" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.136ex; height:2.843ex;" alt="{\\\\displaystyle \\\\|m-x\\\\|\\\\leq \\\\|c-x\\\\|}"></span> for every <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\\\\in C.}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>c</mi>
         <mo>∈<!-- ∈ --></mo>
         <mi>C</mi>
         <mo>.</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle c\\\\in C.}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/925f8320f368db27227d19fae2398599da0e1ee5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.261ex; height:2.176ex;" alt="{\\\\displaystyle c\\\\in C.}">
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Hilbert_projection_theorem">https://en.wikipedia.org/wiki/Hilbert_projection_theorem</a>)"""@en ;
  dc:created "2023-08-17"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_projection_sur_un_convexe_ferm%C3%A9>, <https://en.wikipedia.org/wiki/Hilbert_projection_theorem> ;
  dc:modified "2023-08-17"^^xsd:date ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:related psr:-S7FM9BJ5-N .