Corollary 2

Let $G$ be a locally compact group, $\beta$ a Haar measure on $G$, $E$ a separable Banach space, and $U$ a linear representation of $G$ in $E$ such that $U(s) \in \mathscr{L}(E;E)$ for all $s \in G$. Let $(a_m)$ be a total sequence in $E$, and let $(a'_n)$ be a dense sequence in the unit ball $B'$ of the dual $E'$ of $E$, equipped with the weak topology. Assume that the functions $s \mapsto \langle U(s)a_m, a'_n \rangle$ on $G$ are $\beta$-measurable. Then $U$ is a continuous linear representation.

Let us first show that for every $z' \in E'$, the scalar functions

are $\beta$-measurable; we may restrict ourselves to the case that $\|z'\| \leq 1$, and, since $B'$ is metrizable for the weak topology (TVS, III, §3, No. 4, Cor. 2 of Prop. 6), there exists a subsequence $(a'_{n_k})$ of $(a'_n)$ that converges weakly to $z'$; the function

is thus the limit of a sequence of $\beta$-measurable functions, whence our assertion. It follows that the mapping $s \mapsto U(s)a_m$ of $G$ into $E$ is $\beta$-measurable for every $m$ (Ch. IV, §5, No. 5, Prop. 10). On the other hand, there exists a sequence $(b_m)$ of elements of $E$, linear combinations of the $a_i$, that is dense in the unit ball of $E$. For every $s \in G$, $\|U(s)\| = \sup_m \|U(s)b_m\|$, therefore $s \mapsto \|U(s)\|$ is measurable. Let $K$ be a compact subset of $G$ and let $\varepsilon > 0$. There exists a compact subset $K_0$ of $K$ such that $\beta(K - K_0) \leq \varepsilon$ and such that the restrictions to $K_0$ of the functions $s \mapsto U(s)a_m$ and $s \mapsto \|U(s)\|$ are continuous. Then the $U(s)$ for $s \in K_0$ are equicontinuous, and the topology of pointwise convergence induces on $U(K_0)$ the topology of pointwise convergence in the set of $a_m$ (TVS, III, §3, No. 4, Prop. 5). Consequently the mapping $s \mapsto U(s)$ of $K_0$ into $\mathscr{L}_s(E; E)$ is continuous. It then suffices to apply Cor. 1.