Proposition 7

Let $\varrho$ be a continuous representation of G in a separated locally convex quasi-complete space E. The sum of the finite-dimensional subrepresentations of E is dense in E.

Let $x\in E$ and let U be an open neighbourhood of $x$. The set of measures $\nu \in \mathscr{M}(G)$ such that $\varrho (\nu )x\in U$ is open in $\mathscr{M}(G)$ for the topology of compact convergence (cf. n$^o2$ of V, p. 400). It contains $\varepsilon_e$, hence it contains a measure of the form $\nu =f_1\cdot \mu$ where $f_1\in \mathscr{C}(G)$ (INT, VIII, p. 171, § 4, n$^o7$, Prop. 19) and, consequently, it contains a measure of the form $f_2\cdot \mu$ where $f_2$ is a G-finite function (Prop. 6 of V, p. 464 and Prop. 4 of V, p. 462).

The subrepresentation F of $\boldsymbol{\gamma }_G$ generated by $f_2$ is finite-dimensional. Let $\widetilde{F}$ be the image of the linear mapping $f\mapsto \varrho (f)x$ of F into E. The space $\widetilde{F}$ is finite-dimensional, and it contains the element $\varrho (f_2)x$ of U. Since $\varrho (g)\varrho (f) =\varrho (\boldsymbol{\gamma }_G(g)f)$ for every $g\in G$ and every $f\in F$ (formula (1) of V, p. 406), the space $\widetilde{F}$ is a subrepresentation of $\varrho$ which meets U. The proposition is proved.