@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: a skos:ConceptScheme .
psr:-X77F5QSS-2
  dc:modified "2024-10-18"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Forme_bilin%C3%A9aire>, <https://en.wikipedia.org/wiki/Bilinear_form> ;
  skos:definition """In mathematics, a <b>bilinear form</b> is a bilinear map <span class="texhtml"><i>V</i> × <i>V</i> → <i>K</i></span> on a vector space <span class="texhtml mvar" style="font-style:italic;">V</span> (the elements of which are called <i>vectors</i>) over a field <i>K</i> (the elements of which are called <i>scalars</i>). In other words, a bilinear form is a function <span class="texhtml"><i>B</i> : <i>V</i> × <i>V</i> → <i>K</i></span> that is linear in each argument separately:  <ul><li><span class="texhtml"><i>B</i>(<b>u</b> + <b>v</b>, <b>w</b>) = <i>B</i>(<b>u</b>, <b>w</b>) + <i>B</i>(<b>v</b>, <b>w</b>)</span> <span class="nowrap">   </span> and <span class="nowrap">   </span> <span class="texhtml"><i>B</i>(<i>λ</i><b>u</b>, <b>v</b>) = <i>λB</i>(<b>u</b>, <b>v</b>)</span></li> <li><span class="texhtml"><i>B</i>(<b>u</b>, <b>v</b> + <b>w</b>) = <i>B</i>(<b>u</b>, <b>v</b>) + <i>B</i>(<b>u</b>, <b>w</b>)</span> <span class="nowrap">   </span> and <span class="nowrap">   </span> <span class="texhtml"><i>B</i>(<b>u</b>, <i>λ</i><b>v</b>) = <i>λB</i>(<b>u</b>, <b>v</b>)</span></li></ul> The dot product on <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} ^{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="double-struck">R</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} ^{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {R} ^{n}}"></span> is an example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When <span class="texhtml mvar" style="font-style:italic;">K</span> is the field of complex numbers <span class="texhtml"><b>C</b></span>, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Bilinear_form">https://en.wikipedia.org/wiki/Bilinear_form</a>)"""@en, """En mathématiques, plus précisément en algèbre linéaire, une <b>forme bilinéaire</b> est une application qui à un couple de vecteurs associe un scalaire, et qui a la particularité d'être linéaire en ses deux arguments. Autrement dit, étant donné un espace vectoriel <span class="texhtml mvar" style="font-style:italic;"> <i>V</i></span> sur un corps commutatif <span class="texhtml mvar" style="font-style:italic;"> <i>K</i></span>, il s'agit d'une application <span class="texhtml"><i>f</i> : <i>V</i> × <i>V</i> → <i>K</i></span> telle que, pour tous <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,y,z\\\\in V}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>x</mi>         <mo>,</mo>         <mi>y</mi>         <mo>,</mo>         <mi>z</mi>         <mo>∈<!-- ∈ --></mo>         <mi>V</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle x,y,z\\\\in V}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66794d289d37b67c3857e7370ecdff292a909f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.269ex; height:2.509ex;" alt="{\\\\displaystyle x,y,z\\\\in V}"></span> et tous <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 \\\\alpha ,\\eta \\\\in K}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>α<!-- α --></mi>         <mo>,</mo>         <mi>β<!-- β --></mi>         <mo>∈<!-- ∈ --></mo>         <mi>K</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\alpha ,\\eta \\\\in K}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1016d6171834a060fa8228c16b4d1f845fd3921c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.76ex; height:2.509ex;" alt="{\\\\displaystyle \\\\alpha ,\\eta \\\\in K}"></span>,  <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(\\\\alpha x+\\eta y,\\\\;z)=\\\\alpha f(x,z)+\\eta f(y,z);}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>α<!-- α --></mi>         <mi>x</mi>         <mo>+</mo>         <mi>β<!-- β --></mi>         <mi>y</mi>         <mo>,</mo>         <mspace width="thickmathspace"></mspace>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>α<!-- α --></mi>         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>,</mo>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>+</mo>         <mi>β<!-- β --></mi>         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>y</mi>         <mo>,</mo>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>;</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f(\\\\alpha x+\\eta y,\\\\;z)=\\\\alpha f(x,z)+\\eta f(y,z);}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a779e31a16f04fafce13d58de078ec521fc2b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.311ex; height:2.843ex;" alt="{\\\\displaystyle f(\\\\alpha x+\\eta y,\\\\;z)=\\\\alpha f(x,z)+\\eta f(y,z);}"></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 f(x,\\\\;\\\\alpha y+\\eta z)=\\\\alpha f(x,y)+\\eta f(x,z).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>,</mo>         <mspace width="thickmathspace"></mspace>         <mi>α<!-- α --></mi>         <mi>y</mi>         <mo>+</mo>         <mi>β<!-- β --></mi>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>α<!-- α --></mi>         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>,</mo>         <mi>y</mi>         <mo stretchy="false">)</mo>         <mo>+</mo>         <mi>β<!-- β --></mi>         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>,</mo>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f(x,\\\\;\\\\alpha y+\\eta z)=\\\\alpha f(x,y)+\\eta f(x,z).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cad5159941f66f4256cc441086f9f6ff2c30fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.552ex; height:2.843ex;" alt="{\\\\displaystyle f(x,\\\\;\\\\alpha y+\\eta z)=\\\\alpha f(x,y)+\\eta f(x,z).}"></span></dd></dl> Les formes bilinéaires sont naturellement introduites pour les produits scalaires. Les produits scalaires (sur les espaces vectoriels de dimension finie ou infinie) sont très utilisés, dans toutes les branches mathématiques, pour définir une distance.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Forme_bilin%C3%A9aire">https://fr.wikipedia.org/wiki/Forme_bilin%C3%A9aire</a>)"""@fr ;
  skos:broader psr:-Q8X6082L-Q ;
  skos:inScheme psr: ;
  skos:prefLabel "forme bilinéaire"@fr, "bilinear form"@en ;
  skos:related psr:-VQWWJ37X-0 ;
  a skos:Concept .

psr:-VQWWJ37X-0
  skos:prefLabel "Littlewood's 4/3 inequality"@en, "inégalité 4/3 de Littlewood"@fr ;
  a skos:Concept ;
  skos:related psr:-X77F5QSS-2 .

psr:-Q8X6082L-Q
  skos:prefLabel "forme multilinéaire"@fr, "multilinear form"@en ;
  a skos:Concept ;
  skos:narrower psr:-X77F5QSS-2 .