Exercise 13, § 5

Let E, F, G be three topological vector spaces, and S (resp. T) a family of bounded subsets of E (resp. F). Let H be the vector space of T-hypocontinuous bilinear mappings from E × F into G. a) Show that on H the topology of uniform convergence on sets of the form M × N, where M ∈ S and N ∈ T is compatible with the vector space structure; this topology is called the (S, T)-topology on H. For every mapping u ∈ H, let ũ be the continuous mapping x ↦ u(x, .) from E into L_T(F ; G). Show that u ↦ ũ is an isomorphism from the space H, endowed with the (S, T)-topology, onto the space L_S(E ; L_T(F ; G)). b) Let L be a subset of H such that, for every pair (x, y) ∈ E × F, the set of all u(x, y), where u runs through L, is bounded in G (simply bounded subset of H). Show that, if E, F, G are locally convex, and if E and F are Hausdorff and quasi-complete, then L is bounded in H for the (S, T)-topology. c) Let E, F, G be three Hausdorff locally convex spaces. If E is barrelled and F quasi-complete or barrelled, then every simply bounded subset L of H is T-equihypocontinuous (III, p. 47, exerc. 7). d) If E and F are barrelled, and G quasi-complete, and if S and T are coverings of E and F respectively, then H is Hausdorff and quasi-complete for the (S, T)-topology.