Definition 1

— (i) $U$ is said to be separately continuous if, for every $s \in G$, $U(s)$ is a continuous endomorphism of $E$, and if, for every $x \in E$, the mapping $s \mapsto U(s)x$ of $G$ into $E$ is continuous.

(ii) $U$ is said to be continuous if $(s, x) \mapsto U(s)x$ is a continuous mapping of $G \times E$ into $E$.

(iii) $U$ is said to be equicontinuous if it is continuous and if the set of endomorphisms $U(s)$, where $s$ runs over $G$, is equicontinuous.