@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:-VVTJ8P47-K
  skos:prefLabel "application linéaire"@fr, "linear map"@en ;
  a skos:Concept ;
  skos:narrower psr:-GCF3H53P-P .

psr:-HX2VX066-P
  skos:prefLabel "functional analysis"@en, "analyse fonctionnelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-GCF3H53P-P .

psr:-ZTD7VMDS-3
  skos:prefLabel "analyse convexe"@fr, "convex analysis"@en ;
  a skos:Concept ;
  skos:narrower psr:-GCF3H53P-P .

psr: a skos:ConceptScheme .
psr:-GCF3H53P-P
  dc:modified "2023-08-17"^^xsd:date ;
  skos:broader psr:-ZTD7VMDS-3, psr:-L2BN0W1T-P, psr:-VVTJ8P47-K, psr:-HX2VX066-P ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Sublinear_function>, <https://fr.wikipedia.org/wiki/Application_sous-lin%C3%A9aire> ;
  skos:prefLabel "sublinear function"@en, "application sous-linéaire"@fr ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:definition """Soit <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" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="V"></span> un espace vectoriel sur ℝ. On dit qu'une application <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\\\\,\\\\colon \\\\,V\\	o \\\\mathbb {R} \\\\cup \\\\{+\\\\infty \\\\}}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>s</mi>
         <mspace width="thinmathspace"></mspace>
         <mo>:<!-- : --></mo>
         <mspace width="thinmathspace"></mspace>
         <mi>V</mi>
         <mo stretchy="false">→<!-- → --></mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         <mo>∪<!-- ∪ --></mo>
         <mo fence="false" stretchy="false">{</mo>
         <mo>+</mo>
         <mi mathvariant="normal">∞<!-- ∞ --></mi>
         <mo fence="false" stretchy="false">}</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle s\\\\,\\\\colon \\\\,V\\	o \\\\mathbb {R} \\\\cup \\\\{+\\\\infty \\\\}}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed6ad1a16cd1653a1248aed6a9d42f57d986295" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.017ex; height:2.843ex;" alt="{\\\\displaystyle s\\\\,\\\\colon \\\\,V\\	o \\\\mathbb {R} \\\\cup \\\\{+\\\\infty \\\\}}"></span> est <b>sous-linéaire</b> lorsque :
         </p>
         <ul><li>pour tous vecteurs <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> et <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 y}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>y</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle y}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"></span> de <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" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="V"></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 s(x+y)\\\\leq s(x)+s(y)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>s</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo>+</mo>
         <mi>y</mi>
         <mo stretchy="false">)</mo>
         <mo>≤<!-- ≤ --></mo>
         <mi>s</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo>+</mo>
         <mi>s</mi>
         <mo stretchy="false">(</mo>
         <mi>y</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle s(x+y)\\\\leq s(x)+s(y)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8eecc976f76b3e9dd96c9c296a720e7903cabeb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.449ex; height:2.843ex;" alt="{\\\\displaystyle s(x+y)\\\\leq s(x)+s(y)}"></span> (on dit que <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}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>s</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle s}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="s"></span> est sous-additive),</li>
         <li>pour tout vecteur <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> et tout <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 \\\\lambda \\\\geq 0}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>λ<!-- λ --></mi>
         <mo>≥<!-- ≥ --></mo>
         <mn>0</mn>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lambda \\\\geq 0}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c26004859ae51dde7800b3f3a960c73f81cd583" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.616ex; height:2.343ex;" alt="{\\\\displaystyle \\\\lambda \\\\geq 0}"></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 s(\\\\lambda x)=\\\\lambda \\\\,s(x)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>s</mi>
         <mo stretchy="false">(</mo>
         <mi>λ<!-- λ --></mi>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo>=</mo>
         <mi>λ<!-- λ --></mi>
         <mspace width="thinmathspace"></mspace>
         <mi>s</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle s(\\\\lambda x)=\\\\lambda \\\\,s(x)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5230c11c8b804ae0a5b684f2dfd1b341b4759a48" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.655ex; height:2.843ex;" alt="{\\\\displaystyle s(\\\\lambda x)=\\\\lambda \\\\,s(x)}"></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> (on dit que <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}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>s</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle s}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="s"></span> est positivement homogène).</li> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Application_sous-lin%C3%A9aire">https://fr.wikipedia.org/wiki/Application_sous-lin%C3%A9aire</a>)"""@fr, """In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on 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>
         </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> is a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm except that it is not required to map non-zero vectors to non-zero values. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Sublinear_function">https://en.wikipedia.org/wiki/Sublinear_function</a>)"""@en ;
  dc:created "2023-08-17"^^xsd:date .

psr:-L2BN0W1T-P
  skos:prefLabel "fonction"@fr, "function"@en ;
  a skos:Concept ;
  skos:narrower psr:-GCF3H53P-P .