Proposition 1

Let $\varrho$ be a finite-dimensional linear representation of G in a separated topological vector space E. There exists a scalar product $q$ on E such that $\varrho$ is a unitary representation in the Hilbert space E endowed with $q$. In particular, $\varrho$ is semisimple.

Since E is finite-dimensional, there exists a Hilbert space structure on E. The result follows by applying the preceding remark and corollary 2 of V, p. 392.