Exercise 20, § 4

Let X, Y be two paracompact locally compact spaces. Equip the spaces $\mathscr{C}'(X; \mathbf{C})$ and $\mathscr{C}'(Y; \mathbf{C})$ with the topology of compact convergence. a) Show that when $\mathscr{C}'(X \times Y; \mathbf{C})$ is equipped with the topology of compact convergence, the mapping $(\mu, \nu) \mapsto \mu \otimes \nu$ is $(\mathscr{S}, \mathscr{T})$-hypocontinuous, where $\mathscr{S}$ denotes the set of bounded subsets of $\mathscr{C}'(X; \mathbf{C})$ and $\mathscr{T}$ the set of bounded subsets of $\mathscr{C}'(Y; \mathbf{C})$ (make use of Exer. 18 b)). b) Show that when X and Y are countable at infinity and $\mathscr{C}'(X \times Y; \mathbf{C})$ is equipped with the topology of compact convergence, the mapping $(\mu, \nu) \mapsto \mu \otimes \nu$ is continuous. (Inspired by the proof of Exer. 3 of Ch. III, §4 and introducing in X (resp. Y) a continuous partition of unity subordinate to a locally finite open covering, argue as in TVS, II, §4, Exer. 9 a).) c) If X is discrete, the space $\mathscr{C}'(X; \mathbf{C})$ may be identified with the direct sum space $\mathbf{C}^{(X)}$, equipped with the finest locally convex topology. From this, deduce an example where X and Y are discrete, with X uncountable, and where the mapping $(\mu, \nu) \mapsto \mu \otimes \nu$ is not continuous when $\mathscr{C}'(X \times Y; \mathbf{C})$ is equipped with the vague topology (cf. TVS, II, §4, Exer. 9 b)).