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

psr:-ZMBLSQ5G-X
  skos:definition """Une <b>algèbre normée</b> est une algèbre <i>A</i> sur le corps des réels ou des complexes munie d'une norme d'espace vectoriel qui vérifie&nbsp;:
<br/>
<br/><center><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 \\orall x,y\\\\in A\\\\qquad \\\\|xy\\\\|\\\\leq \\\\|x\\\\|\\\\|y\\\\|.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">∀<!-- ∀ --></mi>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>        <mi>y</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>A</mi>
<br/>        <mspace width="2em"></mspace>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>x</mi>
<br/>        <mi>y</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>y</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\orall x,y\\\\in A\\\\qquad \\\\|xy\\\\|\\\\leq \\\\|x\\\\|\\\\|y\\\\|.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36644184b0ecdf7a6880e591203b12054b63486b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:29.731ex; height:2.843ex;" alt="{\\\\displaystyle \\orall x,y\\\\in A\\\\qquad \\\\|xy\\\\|\\\\leq \\\\|x\\\\|\\\\|y\\\\|.}"></span></center>
<br/>En d'autres termes, il s'agit d'une algèbre sur <i>K</i> = <b>R</b> ou <b>C</b> telle que l'espace vectoriel sous-jacent soit normé, la norme étant en outre sous-multiplicative.
<br/>Dans une algèbre normée unifère <i>A</i> non nulle, l'élément unité peut toujours être supposé de norme 1, quitte à remplacer la norme <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\\\\mapsto \\\\|x\\\\|}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">↦<!-- ↦ --></mo>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x\\\\mapsto \\\\|x\\\\|}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a36029fb96938d2944a2c85f1f2958dcf39c900" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.598ex; height:2.843ex;" alt="{\\\\displaystyle x\\\\mapsto \\\\|x\\\\|}"></span> par la norme équivalente d'algèbre <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\\\\mapsto \\\\sup _{\\\\|y\\\\|=1}\\\\|xy\\\\|}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">↦<!-- ↦ --></mo>
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">sup</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>            <mi>y</mi>
<br/>            <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>x</mi>
<br/>        <mi>y</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x\\\\mapsto \\\\sup _{\\\\|y\\\\|=1}\\\\|xy\\\\|}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef4784c5aaffddf868cee6c2e101b6baca0f8b6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:14.702ex; height:5.009ex;" alt="{\\\\displaystyle x\\\\mapsto \\\\sup _{\\\\|y\\\\|=1}\\\\|xy\\\\|}"></span>.
<br/> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_norm%C3%A9e">https://fr.wikipedia.org/wiki/Alg%C3%A8bre_norm%C3%A9e</a>)"""@fr, """In mathematics, a <b>normed algebra</b> <i>A</i> is an algebra over a field which has a sub-multiplicative norm:
<br/>
<br/><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 \\orall x,y\\\\in A\\\\qquad \\\\|xy\\\\|\\\\leq \\\\|x\\\\|\\\\|y\\\\|.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">∀<!-- ∀ --></mi>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>        <mi>y</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>A</mi>
<br/>        <mspace width="2em"></mspace>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>x</mi>
<br/>        <mi>y</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mi>y</mi>
<br/>        <mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\orall x,y\\\\in A\\\\qquad \\\\|xy\\\\|\\\\leq \\\\|x\\\\|\\\\|y\\\\|.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36644184b0ecdf7a6880e591203b12054b63486b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:29.731ex; height:2.843ex;" alt="{\\\\displaystyle \\orall x,y\\\\in A\\\\qquad \\\\|xy\\\\|\\\\leq \\\\|x\\\\|\\\\|y\\\\|.}"></span></dd></dl>
<br/>Some authors require it to have a multiplicative identity 1<sub><i>A</i></sub> such that ║1<sub><i>A</i></sub>║ = 1. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Normed_algebra">https://en.wikipedia.org/wiki/Normed_algebra</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-CS2FD3K2-0 ;
  skos:prefLabel "algèbre normée"@fr, "normed algebra"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Alg%C3%A8bre_norm%C3%A9e>, <https://en.wikipedia.org/wiki/Normed_algebra> .

psr: a skos:ConceptScheme .
psr:-CS2FD3K2-0
  skos:prefLabel "algebra over a field"@en, "algèbre sur un corps"@fr ;
  a skos:Concept ;
  skos:narrower psr:-ZMBLSQ5G-X .