Proposition 1

— Let E be a locally convex space. The following conditions are equivalent :

(i) Let U be a subset of E which absorbs every bounded subset of E, and which is the intersection of a sequence of convex, balanced and closed neighbourhoods of 0 in E. Then U is a neighbourhood of 0 in E.

(ii) For every locally convex space F, every bounded subset of $\mathscr{L}_b(E; F)$ which is the union of a countable family of equicontinuous subsets, is equicontinuous.

(iii) In the strong dual $E'_b$ of E, every bounded subset which is the union of a countable family of equicontinuous subsets, is equicontinuous.

It is clear that (iii) is a particular case of (ii).

(i) $\Rightarrow$ (ii) : let H be a bounded subset of $\mathscr{L}_b(E; F)$, and let $(H_n)$ be a sequence of equicontinuous subsets of $\mathscr{L}_b(E; F)$ such that $H = \bigcup H_n$. Let V be a convex, balanced and closed neighbourhood of 0 in F. For every $n$, the set $W_n = \bigcap_{u \in H_n} u^{-1}(V)$ is a convex, balanced and closed neighbourhood of 0 in E since $H_n$ is equicontinuous. The set $W = \bigcap_{u \in H} u^{-1}(V)$ absorbs every bounded subset of E, since H is bounded in $\mathscr{L}_b(E; F)$ (III, p. 22), and we have $W = \bigcap_n W_n$. If E satisfies (i), then the set W is a neighbourhood of 0 in E, hence H is equicontinuous.

(iii) $\Rightarrow$ (i) : let $(U_n)$ be a sequence of convex, balanced and closed neighbourhoods of 0 in E. We assume that the set $U = \bigcap_n U_n$ absorbs every bounded subset of E, hence that its polar $U^\circ$ is bounded in $E'_b$. Then the set $B = \bigcup_n U_n^\circ$ is contained in $U^\circ$, hence is bounded in $E'_b$. If E satisfies (iii), the set B is equicontinuous in $E'$; consequently, the polar $B^\circ = \bigcap_n (U_n^\circ)^\circ = \bigcap_n U_n = U$ of B in E is a neighbourhood of 0 in E.