Exercise 4, § 1

Let F and G be two vector spaces in separating duality, G (resp. F) being identified with the dual of F (resp. G) when the latter is assigned the topology σ(F, G) (resp. σ(G, F)). Let $\mathscr{S}$ (resp. $\mathscr{T}$) be a covering of F (resp. G) consisting of convex, balanced and bounded subsets for σ(F, G) (resp. σ(G, F)). Show that the following propositions are equivalent : α) Every set M ∈ $\mathscr{S}$ is precompact for the $\mathscr{T}$-topology. β) Every set N ∈ $\mathscr{T}$ is precompact for the $\mathscr{S}$-topology. γ) On every set M ∈ $\mathscr{S}$, the topology induced by the $\mathscr{T}$-topology is identical with the topology induced by σ(F, G). δ) On every set N ∈ $\mathscr{T}$, the topology induced by the $\mathscr{S}$-topology is identical with the topology induced by σ(G, F). (Use prop. 5 of III, p. 17 to show that α) implies δ) and exerc. 1 of II, p. 74 to show that δ) implies β).)