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.



[a1] G. Warner, "Harmonic analysis on semi-simple Lie groups" , 1 , Springer (1972)
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Continuous representation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article