Proposition 6

When $\mathscr{M}(X; \mathbf{C}) , \ \mathscr{M}(Y; \mathbf{C})$ and $\mathscr{M}(X \times Y; \mathbf{C})$ are equipped with the topology of strictly compact convergence (§ 1, No. 10), the bilinear mapping $(\lambda, \mu) \mapsto \lambda \otimes \mu$ of $\mathscr{M}(X; \mathbf{C}) \times \mathscr{M}(Y; \mathbf{C})$ into $\mathscr{M}(X \times Y; \mathbf{C})$ is hypocontinuous for the set of vaguely bounded subsets of $\mathscr{M}(X; \mathbf{C})$ and $\mathscr{M}(Y; \mathbf{C})$ (TVS, III, § 5, No. 3).

Let $K \subset X , \ L \subset Y$ be two compact sets, $A$ a compact subset of $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$, and $B$ a vaguely bounded and closed subset of $\mathscr{M}(X; \mathbf{C})$; it is known that $B$ is vaguely compact (§ 1, No. 9, Prop. 15), hence also compact for the topology of strictly compact convergence (§ 1, No. 10, Prop. 17). On the other hand, the Banach space $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$ is isometric to $\mathscr{K}(X, K; \mathscr{K}(Y, L; \mathbf{C}))$ (No. 1, Lemma 1); the mapping $\varphi$ of $\mathscr{K}(X, K; \mathscr{K}(Y, L; \mathbf{C})) \times \mathscr{M}(X; \mathbf{C})$ into $\mathscr{K}(Y, L; \mathbf{C})$, such that $\varphi(g, \lambda)$ is the function $h$ defined by $h(y) = \int g(x, y) d\lambda(x)$, is separately continuous by virtue of § 3, No. 4, Props. 8 and 9. Since $\mathscr{K}(X, K; \mathscr{K}(Y, L; \mathbf{C}))$ is barreled, it follows that the mapping $\varphi$ is hypocontinuous relative to the vaguely bounded subsets of $\mathscr{M}(X; \mathbf{C})$ (TVS, III, § 5, No. 3, Prop. 6); the restriction of this mapping to $A \times B$ is therefore continuous (loc. cit., Prop. 4). The image $C$ of $A \times B$ under this mapping is consequently a compact subset of the Banach space $\mathscr{K}(Y, L; \mathbf{C})$. Now, $C$ is none other than the set of functions $h(y) = \int f(x, y) d\lambda(x)$ as $f$ runs over $A$ and $\lambda$ runs over $B$; by virtue of formula (3) of No. 1, the conditions $\lambda \in B$ and $\mu \in C^\circ$ therefore imply $\lambda \otimes \mu \in A^\circ$. In view of the definition of the topology of strictly compact convergence, this proves the proposition (TVS, III, § 5, No. 3, Def. 2).

The conclusion of Prop. 6 is no longer valid when the topology of strictly compact convergence is replaced by the vague topology (Exer. 2 c)). However, if B (resp. C) is a vaguely bounded subset of $\mathscr{M}(X; \mathbf{C})$ (resp. $\mathscr{M}(Y; \mathbf{C})$), then the image of $B \times C$ under the mapping $(\lambda, \mu) \mapsto \lambda \otimes \mu$ is vaguely bounded in $\mathscr{M}(X \times Y; \mathbf{C})$ and therefore the restriction of this mapping to $B \times C$ is vaguely continuous, by virtue of Prop. 6, of §1, No. 10, Prop. 17, and of Prop. 4 of TVS, III, §5, No. 3 (cf. Exer. 3).