Exercise 3, § 4

a) Let $X, Y$ be two locally compact spaces, $K \subset X$, $L \subset Y$ two compact sets, $A$ a compact subset of the space $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$. For every integer $n$, let $V_n$ be an entourage of the uniform structure of $K$ such that, for every pair $(x', x'') \in V_n$, every $y \in L$ and every function $f \in A$,

Let $C_n$ be the set of functions $y \mapsto n(f(x', y) - f(x'', y))$ for all pairs $(x', x'') \in V_n$ and every $f \in A$; show that the union $C$ of the $C_n$ for all $n \geq 1$ is a relatively compact subset of $\mathscr{K}(Y, L; \mathbf{C})$.

b) Let $B$ be the union of $C$ and the set of mappings $y \mapsto f(x, y)$ for $x \in K$ and $f \in A$, which is a relatively compact subset of $\mathscr{K}(Y, L; \mathbf{C})$. Show that when $\nu$ runs over the polar $B^\circ$ of $B$ in $\mathscr{M}(Y; \mathbf{C})$ and $f$ runs over $A$, the set of mappings

is relatively compact in $\mathscr{K}(X, K; \mathbf{C})$.

c) Deduce from b) that the mapping $(\mu, \nu) \mapsto \mu \otimes \nu$ of $\mathscr{M}(X; \mathbf{C}) \times \mathscr{M}(Y; \mathbf{C})$ into $\mathscr{M}(X \times Y; \mathbf{C})$ is continuous 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.