Exercise 18, § 4

Let X be a paracompact locally compact space. a) Show that the space $\mathscr{C}(X; \mathbf{C})$ is isomorphic to a product of Fréchet spaces, hence is barreled. b) For a subset H of $\mathscr{C}'(X; \mathbf{C})$ to be bounded for the topology of compact convergence, it is necessary and sufficient that there exist a compact subset K of X such that $\mathrm{Supp}(\mu) \subset K$ for all $\mu \in H$ and such that H is bounded for the ultrastrong topology (compare with Exer. 17 c)). The topology induced on H by the topology of compact convergence is then identical to the vague topology. c) Show that on the set $\mathscr{M}_+(X) \cap \mathscr{C}'(X; \mathbf{C})$ the topology induced by the topology of compact convergence on $\mathscr{C}'(X; \mathbf{C})$ is identical to the vague topology. Is the same true when X is not paracompact (Exer. 17)?