Lemma 2

Let G be a topological group, U a linear representation of G in a normed space E, and A 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 the function $s \mapsto g(s) = \|U(s)\|$ on G is lower semi-continuous and satisfies $g(st) \leqslant g(s)g(t)$.

Let B be the unit ball of E. Then $g(s) = \sup_{x \in B \cap A} \|U(s)x\|$, and each function $s \mapsto \|U(s)x\|$ is continuous on G, therefore $g$ is lower semi-continuous. On the other hand,