Exercise 19, § 4

Equip $\mathscr{C}(X; \mathbf{C})$ with the topology of compact convergence, and $\mathscr{C}'(X; \mathbf{C})$ with the topology of uniform convergence on the compact subsets of $\mathscr{C}(X; \mathbf{C})$. Let $\mathscr{S}$ be the set of compact subsets of $\mathscr{C}(X; \mathbf{C})$; show that the bilinear mapping $(g, \mu) \mapsto g \cdot \mu$ of $\mathscr{C}(X; \mathbf{C}) \times \mathscr{C}'(X; \mathbf{C})$ into $\mathscr{C}'(X; \mathbf{C})$ is $\mathscr{S}$-hypocontinuous. Let $\mathscr{T}$ be the set of bounded subsets of $\mathscr{C}'(X; \mathbf{C})$; show that if X is paracompact, the preceding bilinear mapping is also $\mathscr{T}$-hypocontinuous; is the latter property again true when X is no longer assumed to be paracompact (Exer. 17)?