Proposition 8

— *Let H be a family of linear mappings from E into a locally convex space F. For H to be equicontinuous, it is necessary and sufficient that for every equicontinuous subset X in the dual F' of F, the set of linear forms f \circ u, for f \in X and u \in H, is equicontinuous.

It is obvious that the condition is necessary. Suppose it is verified, and let V be a closed convex balanced neighbourhood of 0 in F. Since V^\circ is equicontinuous, there exists a neighbourhood W of 0 in E such that |f(u(x))| \leq 1 for all x \in W, u \in H and f \in V^\circ; in other words, u(W) \subset V^{\circ\circ} = V for all u \in H, hence H is equicontinuous.