Proposition 10

*Let $G$ be a locally compact group, $E$ a quasi-complete locally convex space, $U$ a continuous linear representation of $G$ in $E$, and $\beta$ a positive measure on $G$ with support $G$.

(i) The vectors $U(f \cdot \beta)x$, where $f$ runs over $\mathscr{K}(G)$ and $x$ runs over $E$, are dense in $E$.

(ii) Let $F$ be a closed linear subspace of $E$. If $F$ is stable for $U$, then $U(\mu)(F) \subset F$ for every $\mu \in \mathscr{C}'(G)$. Conversely, if $U(f \cdot \beta) \subset F$ for every $f \in \mathscr{K}(G)$, then $F$ is stable for $U$.

The first part of (ii) is immediate, since the restrictions of the $U(s)$ to $F$ ($s \in G$) define a continuous linear representation of $G$ in the quasi-complete locally convex space $F$. The second part of (ii), and (i), follow from Cor. 3 of Lemma 4.