Corollary 2

Let $\varrho$ be a continuous representation of G in a separated quasi-complete locally convex space E and let $\pi$ be an irreducible continuous representation of G.

a) The G-morphism $p_{\pi}=$ dim($\pi$ )$\varrho (\overline{\chi}_{\pi})$ of E into E is a continuous projector of E whose image is the $\pi$-isotypical component of $\varrho$;

b) If $\varrho$ is a unitary representation, then the projector $p_{\pi}$ is the orthoprojector of E with image $M_{\pi}(\varrho )$.

Let us prove a). The linear mapping $p_{\pi}=$ dim($\pi$ )$\varrho (\overline{\chi}_{\pi})$ is well-defined, since E is quasi-complete and G is compact (V, p. 401). It is an element of Hom$_G(\varrho , \varrho )$ since the character of $\pi$ is a continuous central function on G.

Let $\varpi$ be a finite-dimensional subrepresentation of E and F its space. The mapping $p_{\pi}$ induces, by passing to subspaces, the endomorphism (dim $\pi$ )$\varpi (\overline{\chi}_{\pi})$ of F. This endomorphism is therefore zero if $\varpi$ is not isomorphic to $\pi$, and is the identity mapping of F otherwise (in fact, this is the case if $\varpi$ is irreducible by Cor. 5 of V, p. 460, and the general case follows from this since $\varpi$ is semisimple by Prop. 1 of V, p. 457).

Finally, since the sum of the finite-dimensional subrepresentations of E is dense in E (Prop. 7) and $p_{\pi}$ is continuous, it follows that $p_{\pi}$ is a projector whose image is the $\pi$-isotypical component of $\varrho$. If E is Hilbert space, the projector $p_{\pi}$ is hermitian by Cor. 1 of V, p. 458, hence it is an orthoprojector (Lemma 3, (ii) of I, p. 133).