Proposition 9

Let E and F be two locally convex spaces and $\mathscr{S}$ a family of bounded subsets of E. Then every equicontinuous subset of $\mathscr{L}(E; F)$ is bounded for the $\mathscr{S}$-topology.

For, if $H$ is an equicontinuous subset of $\mathscr{L}(E; F)$ and $V$ a neighbourhood of 0 in F, the set $\bigcap_{u \in H} u^{-1}(V)$ is a neighbourhood of 0 in E, hence absorbs every bounded subset of E.

A subset of $\mathscr{L}(E; F)$ which is bounded for a $\mathscr{S}$-topology is not necessarily equicontinuous, even if $\mathscr{S}$ is covering and $\mathscr{S}$ is the canonical bornology on E (IV, p. 50, exerc. 17). In the following paragraph we shall study, under the name barrelled spaces, the spaces E such that every simply bounded subset of $\mathscr{L}(E; F)$ is equicontinuous. For the present note the following result :