Lemma 4

Let G be a topological group, and let $\varrho$ be a homomorphism of G into the unitary group $\mathbf{U}(E)$ of a Hilbert space E. Then $\varrho$ is a unitary representation of G if and only if $\varrho$ is continuous at the element $e$ of G for the topology of simple convergence on $\mathbf{U}(E)$. It is enough that this property be valid for every $x$ in a total subset of E.

The condition is obviously necessary. It is sufficient by Remark 2 of INT, VIII, p. 129, § 2, n$^o1$, since the image of $\varrho$ is equicontinuous in $\mathscr{L}(E)$ and since the continuity of the mapping $g\mapsto \varrho (g)x$ at $e$ implies its continuity on G.

Let G be a topological group. If $\varrho_1$ and $\varrho_2$ are unitary representations of G and if $u$ belong to Hom$_G(\varrho_1, \varrho_2)$, then $u^*$ belong to Hom$_G(\varrho_2, \varrho_1)$. In fact, since $\varrho_1$ and $\varrho_2$ are unitary representations, one has for every $g\in G$