Proposition 1

Let H be a locally compact group. Let $\varrho_1$ (resp. $\varrho_2$) be a continuous representation of G (resp. of H) in a separated locally convex K-vector space $E_1$ (resp. $E_2$). Let F denote the space $\mathscr{L}(E_1; E_2)$ endowed with the topology of compact convergence. The representation $\varrho$ of $G\times H$ in F defined by $\varrho (g, h)u=\varrho_2(h)\circ u\circ \varrho_1(g^{-1})$ for $(g, h)\in G\times H$ is continuous.

Let $\mathscr{L}_c(E_1; E_2)$ denote the space $\mathscr{L}(E_1; E_2)$ endowed with the topology of compact convergence. Let $\mathfrak{F}_1$ (resp. $\mathfrak{F}_2,\mathfrak{F}_3$) be a filter in G converging to $e$ (resp. a filter in $\mathscr{L}_c(E_1; E_2)$ converging to 0, a filter in H converging to $e$). Since G and H are locally compact, there exist elements $C\in \mathfrak{F}_1$ and $D\in \mathfrak{F}_3$ which are relatively compact. The set $\varrho_1(C^{-1})$ is equicontinuous in $\mathscr{L}(E_1) ($cf. INT, VIII, p. 129, § 2, n$^o1$, rem. 2, a$'))$. By prop. 9 of EVT, III, p. 33 and prop. 4 of EVT, III, p. 31, the mapping defined by $(u, v)\mapsto u\circ v$ of $\mathscr{L}_c(E_1; E_2)\times \varrho_1(C^{-1})$ into $\mathscr{L}_c(E_1; E_2)$ is continuous. The filter base $\mathfrak{F}_2\circ \varrho_1(\mathfrak{F}^{-1}_1)$ therefore converges to 0 in $\mathscr{L}_c(E_1; E_2)$. The set $\varrho_2(D)$ is equicontinuous in $\mathscr{L}(E_2) ($cf. INT, VIII, p. 129, § 2, n$^o1$, rem. 2), and for every $x\in E_2$, the set $\varrho_2(D)x\subset E_2$ is relatively compact. Consequently, $\varrho_2(D)$ is relatively compact in $\mathscr{L}(E_2)$ endowed with the topology of compact convergence (TG, X, p. 18, cor. 1). The mapping defined by $(u, v)\mapsto u\circ v$ of $\overline{\varrho_2(D)}\times \mathscr{L}_c(E_1; E_2)$ into $\mathscr{L}_c(E_1; E_2)$ is continuous by prop. 9 of EVT, III, p. 33 and prop. 4 of EVT, III, p. 31. Hence the filter base $\varrho (\mathfrak{F}_1\times \mathfrak{F}_3)(\mathfrak{F}_2)$ converges to 0 in $\mathscr{L}_c(E_1; E_2)$. This implies the assertion.