Corollary 2

Let $\varrho$ and $\pi$ be irreducible unitary representations of a topological group G in Hilbert spaces E and F respectively. The space Hom$_G(\varrho , \pi )$ is of dimension 1 if $\varrho$ is isomorphic to $\pi$, and is zero otherwise. In particular, if $\varrho$ is isomorphic to $\pi$, every nonzero G-morphism of $\varrho$ into $\pi$ is an isomorphism.

Suppose there exists a nonzero G-morphism $u$ of $\varrho$ into $\pi$. The linear mapping $u^*\circ u$ is an element of Hom$_G(\varrho , \varrho )$, hence there exists a complex number $\lambda$ such that $u^*\circ u=\lambda \cdot 1_E$ (Prop. 6). We then have

for all $x$ and $y$ in E. In particular, $\lambda \not = 0$ since $u$ is nonzero. Since $\|u(x)\|=|\lambda |^{1/2}\|x\|$ for all $x\in E$, the linear mapping $u$ is injective, and the image of $u$ is closed in F (Lemma 8 of I, p. 107); it is therefore a nonzero subrepresentation of the irreducible representation $\pi$, whence one deduces that $u$ is surjective. Thus, $u$ is an isomorphism of $\varrho$ onto $\pi$. The mapping $v\mapsto u^*\circ v$ is then an isomorphism of the space Hom$_G(\varrho , \pi )$ onto the space Hom$_G(\varrho , \varrho )$, which is of dimension 1 (Prop. 6).