Exercise 2, § 2

Let $U_1$ (resp. $U_2$) be a continuous linear representation of a locally compact group $G$ (resp. $H$) in a locally convex space $E$ (resp. $F$). For $u \in \mathscr{L}(E; F)$, $x \in G$, $y \in H$, set

Show that the mapping $(x, y) \mapsto V(x, y)$ is a continuous linear representation of the group $G^0 \times H$ in the space $\mathscr{L}(E; F)$ equipped with the topology of compact convergence. (Use Prop. 9 of TVS, III, §4, No. 4 and the fact that, for K compact in $G$, $U_1(K)$ is equicontinuous.)