Proposition 10

Let E be a bornological space (in particular, a metrizable locally convex space) and F a locally convex space. Every subset H of $\mathscr{L}(E; F)$ which is bounded for the topology of bounded convergence is equicontinuous.

For every convex balanced neighbourhood $V$ of 0 in F, the set $\bigcap_{u \in H} u^{-1}(V)$ absorbs every bounded subset of E, hence is a neighbourhood of 0 in E ; this proves that H is equicontinuous.