Proposition 2

Let G be a locally compact group, U a linear representation of G in a normed space E. Let A be a dense subset of E. Assume that for every $s \in G$, $U(s)$ is continuous and that, for every $x \in A$, $s \mapsto U(s)x$ is a continuous mapping of G into E. Then U is continuous.

For, $\|U(s)\|$ is bounded on every compact subset of G by Lemmas 1 and 2, and one then applies Remark 2.