Corollary 4

Let $\varrho$ be a unitary representation of G in a Hilbert space E. The space E is the Hilbert sum of the $\pi$-isotypical components $M_{\pi}(\varrho )$ for $\pi \in \widehat{G}$. In particular, the representation $\varrho$ is semisimple.

The isotypical components of $\varrho$ corresponding to non-isomorphic irreducible representations of G are orthogonal (prop. 8 of V, p. 394). Since every finite-dimensional unitary representation of G is semisimple (prop. 1 of V, p. 457), the sum of the isotypical components $M_{\pi}(\varrho )$ for $\pi \in \widehat{G}$ is dense in E (prop. 7). The corollary follows.