# Continuous 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.