Proposition 3

Let $\varrho$ be a unitary representation of a topological group G in a Hilbert space E. Let $\pi$ be a subrepresentation of E and F the space of $\pi$. The orthogonal complement $F^{\circ}$ of F in E is a subrepresentation of E such that $E = F\oplus F^{\circ}$ and $F^{\circ}$ is isomorphic to $E/F$.

The space $F^{\circ}$ is closed. For every $x\in F^{\circ}$, every $g\in G$ and every $y\in F$, one has $\langle \varrho (g)x|y\rangle =\langle x|\varrho (g^{-1})y\rangle = 0$ since $\varrho$ is unitary and F is a subrepresentation. Hence $\varrho (g)x\in F^{\circ}$.

The canonical projection of E onto $E/F$ is a G-morphism, hence the mapping of $F^{\circ}$ into $E/F$ which is deduced from it by passing to the subspace is a G-morphism of $F^{\circ}$ into $E/F$; by EVT, V, p. 13, it is an isometric isomorphism.

We shall also denote by $\pi^{\circ}$ the representation of G in $F^{\circ}$.