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Difference between revisions of "Continuous representation"

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A linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257301.png" /> of a topological group (semi-group, algebra) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257302.png" /> in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257303.png" /> such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257305.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257306.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c0257309.png" />, is continuous. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c02573010.png" /> is continuous in each argument separately, then in certain cases (for example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c02573011.png" /> is a locally compact group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c02573012.png" /> is a Banach space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c02573013.png" /> is automatically continuous jointly in the arguments, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025730/c02573014.png" /> is a continuous representation.
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A linear representation $\pi$ of a topological group (semi-group, algebra) $X$ in a topological vector space $E$ such that the mapping $\phi$ of $E \times X$ into $E$ defined by the formula $\phi(\xi,x) = \pi(x)\xi$, $\xi \in E$, $x \in X$, is continuous. If $\phi$ is continuous in each argument separately, then in certain cases (for example, when $X$ is a locally compact group and $E$ is a Banach space) $\phi$ is automatically continuous jointly in the arguments, that is, $\pi$ is a continuous representation.
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Warner,  "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Warner,  "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)</TD></TR>
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</table>
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Latest revision as of 20:40, 25 October 2014

A linear representation $\pi$ of a topological group (semi-group, algebra) $X$ in a topological vector space $E$ such that the mapping $\phi$ of $E \times X$ into $E$ defined by the formula $\phi(\xi,x) = \pi(x)\xi$, $\xi \in E$, $x \in X$, is continuous. If $\phi$ is continuous in each argument separately, then in certain cases (for example, when $X$ is a locally compact group and $E$ is a Banach space) $\phi$ is automatically continuous jointly in the arguments, that is, $\pi$ is a continuous representation.


Comments

References

[a1] G. Warner, "Harmonic analysis on semi-simple Lie groups" , 1 , Springer (1972)
How to Cite This Entry:
Continuous representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_representation&oldid=16881
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article