Proposition 16

— Let $G, H$ be two Hausdorff locally convex spaces, with $H$ assumed to be quasi-complete. Let $U$ be a mapping of $T$ into $\mathscr{L}(G; H)$ such that $(t, x) \mapsto U(t) \cdot x$ is a continuous mapping of $T \times G$ into $H$. Then the bilinear mapping $(\lambda, x) \mapsto U(\lambda) \cdot x$ of $\mathscr{C}'(T) \times G$ into $H$ is hypocontinuous relative to the equicontinuous subsets of $\mathscr{C}'(T)$ and the compact subsets of $G$ (which implies that the linear mapping $\lambda \mapsto U(\lambda)$ of $\mathscr{C}'(T)$ into $F$ is continuous).

The continuity of $\lambda \mapsto U(t)$ as a mapping of $\mathscr{C}'(T)$ into $F$ follows from Lemma 3 and Remark 2 following Prop. 14 of No. 6. Thus, it remains to prove that for every closed, balanced, convex neighborhood $V$ of 0 in $H$ and for every equicontinuous subset $N$ of $\mathscr{C}'(T)$, there exists a neighborhood $W$ of 0 in $G$ such that the relations $x \in W, \lambda \in N$ imply that $U(\lambda) \cdot x \in V$. One can suppose that $N = S^\circ$, where $S$ is a neighborhood of 0 in $\mathscr{C}(T)$, consequently one can suppose that $S$ is the set of functions $g \in \mathscr{C}(T)$ such that $|g(t)| \leq 1$ on a compact subset $K$ of $T$. It suffices to show that

|⟨U(λ) · x, x′⟩| ≤ 1 for x ∈ W, x′ ∈ V° and λ ∈ S°. Now, since U(K) is equicontinuous, there exists a neighborhood W of 0 in G such that the relations t ∈ K, x ∈ W imply U(t) · x ∈ V; the relations x ∈ W, x′ ∈ V° therefore imply that the function t ↦ ⟨U(t) · x, x′⟩ belongs to S, hence that |⟨U(t) · x, x′⟩| = |∫⟨U(t) · x, x′⟩ dλ(t)| ≤ 1 by the definition of S°.

Let us now assume that U is a continuous mapping of T into F and, moreover, that U(T) is equicontinuous. Then, the same reasoning as above shows (since H is quasi-complete) that for every bounded positive measure μ on T, ∫ U dμ ∈ F. One can therefore define, as above, a linear mapping λ ↦ ∫ U dλ = U(λ) of M^1(T) into F that extends the analogous mapping of C'(T) into F. Moreover, for every closed, balanced, convex neighborhood V of 0 in H, there exists by hypothesis a neighborhood W of 0 in G such that for every x ∈ W and every t ∈ T, one has U(t) · x ∈ V, consequently (since V is weakly closed) ∫ (U(t) · x)dλ(t) ∈ ||λ|| · V (No. 2, Prop. 5). In other words: