Lemma 3

Let $G, H$ be two locally convex spaces, $U$ a mapping of a locally compact space $T$ into $\mathscr{L}(G; H)$. The following conditions are equivalent: a) The mapping $(t, x) \mapsto U(t) \cdot x$ of $T \times G$ into $H$ is continuous. b) For every compact subset $K$ of $T$, $U(K)$ is equicontinuous, and there exists a total set $D \subset G$ such that for every $x \in D$, the mapping $t \mapsto U(t) \cdot x$ is continuous on $T$.

Moreover, when $U$ satisfies these conditions, $U$ is a continuous mapping of $T$ into $\mathscr{L}(G; H)$ equipped with the topology of compact convergence.

To see that a) implies b), we observe that for every neighborhood $V$ of 0 in $H$ and every $t \in K$, there exist by hypothesis a neighborhood $L_t$ of $t$ in $T$ and a neighborhood $W_t$ of 0 in $G$ such that the relations $t' \in L_t$ and $x \in W_t$ imply $U(t') \cdot x \in V$. It suffices to cover $K$ with a finite number of neighborhoods $L_{t_i}$ and to take $W = \bigcap_i W_{t_i}$ in order to have $U(t) \cdot x \in V$ whenever $t \in K$ and $x \in W$, which proves the equicontinuity of $U(K)$.

Conversely, suppose b) is verified; it suffices to show that for every compact subset $K$ of $T$, the mapping $(t, x) \mapsto U(t) \cdot x$ is continuous on $K \times G$. Let $M = U(K)$; since $M$ is equicontinuous, it follows that on $M$, the topology of pointwise convergence in $G$ is identical to the topology of pointwise convergence in $D$ (GT, X, §2, No. 4, Th. 1); the hypothesis b) therefore implies that $t \mapsto U(t)$ is a continuous mapping of $K$ into $\mathscr{L}(G; H)$ when $\mathscr{L}(G; H)$ is equipped with the topology of pointwise convergence. On the other hand, $(A, x) \mapsto A \cdot x$ is a continuous mapping of $M \times G$ into $H$ when $M$ is equipped with the topology of pointwise convergence (GT, X, §2, No. 1, Cor. 4 of Prop. 1). Since the mapping $(t, x) \mapsto U(t) \cdot x$ may be factored as $(t, x) \mapsto (U(t), x) \mapsto U(t) \cdot x$, we conclude that it is continuous.

Finally, the last assertion of the lemma follows from the fact that, on $M$, the topology of compact convergence is identical to that of pointwise convergence (GT, X, §2, No. 4, Th. 1).

Thus, suppose that $U$ satisfies the conditions of Lemma 3; then (if $H$ is quasi-complete) one defines, as in No. 6, a linear mapping $\lambda \mapsto \int U d\lambda$ of $\mathscr{C}'(T)$ into $F = \mathscr{L}(G; H)$. We set $U(\lambda) = \int U d\lambda$.