Exercise 10, § 2

Let G be a locally compact group, E a Hausdorff locally convex space, $E'$ its dual, and $U$ a linear representation of G in E, continuous for the weakened topology $\sigma(E, E')$ on E. Assume that E is quasi-complete for $\sigma(E, E')$, so that $U(\mu)$ is defined for every measure $\mu \in \mathscr{C}'(G)$. Show that the bilinear mapping $(\mu, x) \mapsto U(\mu) \cdot x$ is hypocontinuous relative to the equicontinuous subsets of $\mathscr{C}'(G)$.