Proposition 17

On the space $\mathscr{M}(X; \mathbf{C})$, consider the following topologies: $\mathscr{T}_1$: the topology of pointwise convergence in a total subset T of $\mathscr{K}(X; \mathbf{C})$; $\mathscr{T}_2$: the vague topology; $\mathscr{T}_3$: the topology of strictly compact convergence; $\mathscr{T}_4$: the topology of compact convergence. Each of these topologies is coarser than the next. Moreover: (i) The bounded sets are the same for $\mathscr{T}_2$, $\mathscr{T}_3$ and $\mathscr{T}_4$. (ii) If H is a vaguely bounded subset of $\mathscr{M}(X; \mathbf{C})$, the topologies induced on H by the topologies $\mathscr{T}_1$, $\mathscr{T}_2$, $\mathscr{T}_3$, $\mathscr{T}_4$ are identical.

A vaguely bounded subset $H$ of $\mathscr{M}(X; \mathbf{C})$ is equicontinuous (No. 9, Prop. 15), thus the first assertion follows from TVS, III, §3, No. 7, Prop. 9, and the second from GT, X, §2, No. 4, Th. 1.

Recall that when $X$ is paracompact, the topology of strictly compact convergence coincides with the topology of compact convergence (No. 1, Prop. 2).