Corollary

Let G be a topological group and $\varrho$ a unitary representation of G in a complex Hilbert space E. The Hilbert sum of the $\pi$-isotypical components of G for $\pi \in \widehat{G}$ is the greatest semisimple subrepresentation of $\varrho$.

In fact, the $\pi$-isotypical components of G for $\pi \in \widehat{G}$ are pairwise orthogonal (prop. 8, a)), and each is semisimple (prop. 10, a)); hence the Hilbert sum F of the spaces $M_{\pi}(\varrho )$ for $\pi \in \widehat{G}$ defines a semisimple subrepresentation of $\varrho$. Since moreover every irreducible subrepresentation of $\varrho$ is a subrepresentation of an isotypical component of $\varrho$, the corollary follows.