Exercise 12, § 5

Let E, F be two topological vector spaces, and let f be the bilinear mapping (x, u) ↦ u(x) from E × L(E ; F) into F; let T be a topology compatible with the vector space structure of L(E ; F) and finer than the topology of simple convergence. Let S be a family of bounded subsets of E, U a family of bounded subsets of L(E ; F) (for the topology T). Show that f is S-hypocontinuous if and only if T is finer than the S-topology; f is U-hypocontinuous if and only if the sets of U are equicontinuous subsets of L(E ; F).