Exercise 5, § 2

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$, and $U$ a continuous linear representation of $H$ in a locally convex space $E$. Let $K$ be a compact subset of $G$. Let $\mathscr{K}^U(K)$ be the space of continuous functions on $G$, with values in $E$, with support contained in $KH$, and satisfying $f(xh) = U(h)^{-1}f(x)$ ($x \in G, h \in H$). Let $\mathscr{K}^U$ be the union of the $\mathscr{K}^U(K)$, equipped with the direct limit topology of the topologies of uniform convergence in $K$ on each of the spaces $\mathscr{K}^U(K)$. For $f \in \mathscr{K}^U$ and $s \in G$, define $V(s)f \in \mathscr{K}^U$ by

Show that $V$ is a continuous linear representation of $G$ in $\mathscr{K}^U$.