# Difference between revisions of "Continuous representation"

From Encyclopedia of Mathematics

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− | A linear representation | + | 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> | + | <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> | ||

+ | |||

+ | {{TEX|done}} |

## 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