Exercise 3, § 2

Let $G$ be a locally compact group, $U$ a continuous linear representation of $G$ in a locally convex space $E$, and $E'$ the dual of $E$ equipped with the strong topology. a) Show that for every compact subset $K$ of $G$, $^tU(K)$ is equicontinuous. b) Let $F$ be the set of $a' \in E'$ such that the mapping $s \mapsto ^tU(s)a'$ of $G$ into $E'$ is continuous. Show that $F$ is a closed linear subspace of $E'$ stable for $^tU(G)$ and that the representation deduced by restricting to $F$ the contragredient representation of $U$ is continuous. c) Assume that $E$ is quasi-complete. Let $\alpha$ be a left Haar measure on $G$. Show that $f \mapsto U(f \cdot \alpha)$ is a continuous mapping of $\mathscr{K}(G)$ into $\mathscr{L}(E; E)$ equipped with the topology of bounded convergence. (Use Prop. 17 of Ch. VI, §1, No. 7.) Show that $F$ is weakly dense in $E'$. (Prove that $^tU(f)a' \in F$ for all $a' \in E'$ and $f \in \mathscr{K}(G)$, then use Cor. 3 of Lemma 4.) Deduce from this that if $E$ is semi-reflexive, then the contragredient representation of $U$ in $E'$ equipped with the strong topology is continuous. d) Show that if $U$ is taken to be the left regular representation of $G$ in $L^1(G, \alpha)$ ($\alpha$ still being a left Haar measure on $G$), then $F$ is the subspace of $E' = L^\infty(G, \alpha)$ formed by the uniformly continuous functions.