Corollary 3

Let $\varrho$ be a continuous representation of G in a separated quasi-complete locally convex space E. The endomorphism

of E is a projector of E whose image is the space $E^G$ of invariant vectors in E. In particular, if E is finite-dimensional, then

dim(E$^G$) $=\int_G\chi_{\varrho}(g)d\mu(g)$.

The first assertion is the particular case of the preceding corollary when $\pi$ is the trivial representation of dimension 1 of G. The second follows from it since the dimension of $E^G$ is the trace of the orthoprojector onto $E^G$, that is,

dim(E$^G$) $=$ Tr$(\int_G\varrho (g)d\mu(g))=\int_G\chi_{\varrho}(g)d\mu(g)$.

In particular, when E is finite-dimensional, there exists a vector $x\not = 0$ in E such that $\varrho (g)x=x$ for every $g\in G$ if and only if