Proposition 2

Let $\varrho$ be a unitary representation of a topological group G in a Hilbert space E. Let $u$ denote the isometric isomorphism of $\overline{E}$ onto $E'$ which associates to $x$ the linear form $y\mapsto \langle x|y\rangle$.

a) The mapping $u$ is an isomorphism of the conjugate representation $\overline{\varrho}$ onto the contragredient representation $\breve{\varrho}$;

b) Endow $E'$ with the structure of a Hilbert space obtained by transport of structure by means of $u$; the contragredient representation $\breve{\varrho}$ is a unitary representation of G in $E'$.

By EVT, V, p. 15, Theorem 3 and the following remark, the mapping $u$ is an isometric isomorphism.

For every $g$ in G, every $x$ in $\overline{E}$ and every $y$ in E, one has

hence $\breve{\varrho}(g)\circ u=u\circ \overline{\varrho}(g)$, which proves a); assertion b) follows immediately.