@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-W0JJX1W8-X
  skos:prefLabel "vector space"@en, "espace vectoriel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-CF9WKGJL-W .

psr: a skos:ConceptScheme .
psr:-CF9WKGJL-W
  skos:definition """En mathématiques, un espace hermitien est un espace vectoriel sur le corps commutatif des complexes de dimension finie et muni d'un produit scalaire hermitien. La géométrie d'un tel espace est analogue à celle d'un espace euclidien. De nombreuses propriétés sont communes aux deux structures. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Espace_hermitien">https://fr.wikipedia.org/wiki/Espace_hermitien</a>)"""@fr, """A complex <b>Hermitian form</b> (also called a <b>symmetric sesquilinear form</b>), is a sesquilinear form <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:V\\	imes V\\	o \\\\mathbb {C} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>h</mi>
<br/>        <mo>:</mo>
<br/>        <mi>V</mi>
<br/>        <mo>×<!-- × --></mo>
<br/>        <mi>V</mi>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">C</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle h:V\\	imes V\\	o \\\\mathbb {C} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd723fdb3c6966980bfdc261b495ef840ae6704" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:14.983ex; height:2.176ex;" alt="{\\\\displaystyle h:V\\	imes V\\	o \\\\mathbb {C} }"></span> such that
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle h(w,z)={\\\\overline {h(z,w)}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>h</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>w</mi>
<br/>        <mo>,</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mover>
<br/>            <mrow>
<br/>              <mi>h</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>z</mi>
<br/>              <mo>,</mo>
<br/>              <mi>w</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mo accent="false">¯<!-- ¯ --></mo>
<br/>          </mover>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle h(w,z)={\\\\overline {h(z,w)}}.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25413f079285528a40a10311c32db37d37a7472" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:17.729ex; height:3.676ex;" alt="{\\\\displaystyle h(w,z)={\\\\overline {h(z,w)}}.}"></div>
<br/>The standard Hermitian form 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 {C} ^{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">C</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {C} ^{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {C} ^{n}}"></span> is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\langle w,z\\
angle =\\\\sum _{i=1}^{n}{\\\\overline {w}}_{i}z_{i}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<br/>        <mi>w</mi>
<br/>        <mo>,</mo>
<br/>        <mi>z</mi>
<br/>        <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>w</mi>
<br/>              <mo accent="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\langle w,z\\
angle =\\\\sum _{i=1}^{n}{\\\\overline {w}}_{i}z_{i}.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/068c328b1030876cc80b53dcbeac12344c565426" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -3.005ex; width:17.543ex; height:6.843ex;" alt="{\\\\displaystyle \\\\langle w,z\\
angle =\\\\sum _{i=1}^{n}{\\\\overline {w}}_{i}z_{i}.}"></div>
<br/>More generally, the inner product on any complex Hilbert space is a Hermitian form.
<br/>A minus sign is introduced in the Hermitian form <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 ww^{*}-zz^{*}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>w</mi>
<br/>        <msup>
<br/>          <mi>w</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>z</mi>
<br/>        <msup>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle ww^{*}-zz^{*}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4154ee5dc62489869bf087baf74ac99caacc7a55" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:10.456ex; height:2.509ex;" alt="{\\\\displaystyle ww^{*}-zz^{*}}"></span> to define the group SU(1,1).
<br/>A vector space with a Hermitian form <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,h)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>V</mi>
<br/>        <mo>,</mo>
<br/>        <mi>h</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (V,h)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3217d450cf010fcd93b3284c557e6d20f764d19" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.969ex; height:2.843ex;" alt="{\\\\displaystyle (V,h)}"></span> is called a <b>Hermitian space</b>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form">https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-W0JJX1W8-X ;
  skos:prefLabel "Hermitian space"@en, "espace hermitien"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Espace_hermitien>, <https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form> .