Proposition 4

The linear representations $\gamma_\chi$ of G in $C(X)$ and $K(X)$ are continuous.

The mapping $(s, f) \mapsto (s, \gamma(s)f)$ of $G \times C(X)$ into $G \times C(X)$ is continuous (No. 2, Lemma 3). On the other hand, the mapping $(s, f) \mapsto \chi(s, \cdot)f$ of $G \times C(X)$ into $C(X)$ is continuous; for, if s tends to $s_0$ in G, then $\chi(s, \cdot)$ tends to $\chi(s_0, \cdot)$ uniformly on every compact subset of X; if, moreover, $f$ tends to $f_0$ in $C(X)$, then $\chi(s, \cdot)f$ tends to $\chi(s_0, \cdot)f_0$ uniformly on every compact subset of X, whence our assertion. Thus the representation $\gamma_\chi$ of G in $C(X)$ is continuous.

Let us show that the representation $\gamma_\chi$ of G in $K(X)$ is continuous. Since $K(X)$ is the direct limit of Banach spaces, it is barreled (TVS, III, §4, No. 1, Cor. 3 of Prop. 3), thus it suffices to prove that $\gamma_\chi$ is separately continuous (No. 1, Prop. 1). Now, let H be a compact subset of X and let $s_0 \in G$. Let V be a compact neighborhood of $s_0$ in G, and let L = VH, which is compact in X. For every $f \in K(X, H)$, the support of $\gamma_\chi(s_0)f$ is contained in L, and

therefore $f \mapsto \gamma_\chi(s_0)f$ is a continuous linear mapping of $K(X, H)$ into $K(X, L)$; it follows that $f \mapsto \gamma_\chi(s_0)f$ is a continuous linear mapping of $K(X)$ into itself (TVS, II, §4, No. 4, Prop. 5). On the other hand, the topology of $K(X, L)$ is induced by that of $C(X)$. By what has already been proved, the mapping $s \mapsto \gamma_\chi(s)f$ of V into $K(X, L)$ is continuous. This completes the proof that $\gamma_\chi$ is separately continuous.