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Difference between revisions of "Complete set of functionals"

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''total set of functionals''
 
''total set of functionals''
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023870/c0238701.png" /> of continuous linear functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023870/c0238702.png" />, defined on a linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023870/c0238703.png" />, such that there is no element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023870/c0238704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023870/c0238705.png" />, on which the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023870/c0238706.png" /> is satisfied for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023870/c0238707.png" />. Every locally convex space has a complete set of functionals.
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A set $\Gamma$ of continuous linear functionals $f(x)$, defined on a linear topological space $X$, such that there is no element $x\in X$, $x\neq0$, on which the equality $f(x)=0$ is satisfied for all $f\in\Gamma$. Every locally convex space has a complete set of functionals.

Latest revision as of 15:01, 8 August 2014

total set of functionals

A set $\Gamma$ of continuous linear functionals $f(x)$, defined on a linear topological space $X$, such that there is no element $x\in X$, $x\neq0$, on which the equality $f(x)=0$ is satisfied for all $f\in\Gamma$. Every locally convex space has a complete set of functionals.

How to Cite This Entry:
Complete set of functionals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_set_of_functionals&oldid=16190
This article was adapted from an original article by M.I. Kadets (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article