@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:-W3MQGWL9-8
  skos:prefLabel "polar set"@en, "ensemble polaire"@fr ;
  skos:broader psr:-HX2VX066-P, psr:-ZTD7VMDS-3, psr:-RZ3QL167-D ;
  dc:modified "2023-08-16"^^xsd:date ;
  a skos:Concept ;
  skos:definition """In functional and convex analysis, and related disciplines of mathematics, the <b>polar set</b> <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^{\\\\circ }}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>A</mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mo>∘<!-- ∘ --></mo>
         </mrow>
         </msup>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle A^{\\\\circ }}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee7361a1f050d55a3c14f2c8ce53e9a4f5f6fc0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\\\\displaystyle A^{\\\\circ }}"></span> is a special convex set associated to any subset <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" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> of a vector 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 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/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="X,"></span> lying in the dual 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 X^{\\\\prime }.}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>X</mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
         </mrow>
         </msup>
         <mo>.</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle X^{\\\\prime }.}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/980210d5ccf78c678264901dc7b0ce8a53d827bc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.328ex; height:2.509ex;" alt="{\\\\displaystyle X^{\\\\prime }.}"></span> 
         The <b>bipolar</b> of a subset is the polar of <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^{\\\\circ },}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>A</mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mo>∘<!-- ∘ --></mo>
         </mrow>
         </msup>
         <mo>,</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle A^{\\\\circ },}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ea377173d4c6d133797c7aae9516cce10a85a2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.444ex; height:2.676ex;" alt="{\\\\displaystyle A^{\\\\circ },}"></span> but lies 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 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" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> (not <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^{\\\\prime \\\\prime }}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>X</mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
         <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
         </mrow>
         </msup>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle X^{\\\\prime \\\\prime }}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f905bf6c4a877c966241e9d7650d10e3978b3dc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.134ex; height:2.509ex;" alt="{\\\\displaystyle X^{\\\\prime \\\\prime }}"></span>).
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Polar_set">https://en.wikipedia.org/wiki/Polar_set</a>)"""@en, """En analyse fonctionnelle et en analyse convexe, le polaire d'une partie <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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"></span> d'un espace localement convexe est un convexe fermé de son dual topologique, contenant l'origine et ayant une « relation de dualité » avec <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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"></span>. Bien qu'il soit usuellement défini dans le cadre bien plus général de deux espaces en dualité, nous nous limiterons dans cet article au cas d'un espace euclidien, qui s'identifie à son dual. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Ensemble_polaire">https://fr.wikipedia.org/wiki/Ensemble_polaire</a>)"""@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Polar_set>, <https://fr.wikipedia.org/wiki/Ensemble_polaire> ;
  skos:inScheme psr: ;
  skos:altLabel "polaire"@fr ;
  dc:created "2023-08-16"^^xsd:date .

psr:-RZ3QL167-D
  skos:prefLabel "espace vectoriel topologique"@fr, "topological vector space"@en ;
  a skos:Concept ;
  skos:narrower psr:-W3MQGWL9-8 .

psr:-HX2VX066-P
  skos:prefLabel "functional analysis"@en, "analyse fonctionnelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-W3MQGWL9-8 .

psr:-ZTD7VMDS-3
  skos:prefLabel "analyse convexe"@fr, "convex analysis"@en ;
  a skos:Concept ;
  skos:narrower psr:-W3MQGWL9-8 .

psr: a skos:ConceptScheme .