Exercise 3, § 5

a) Let E be a Hausdorff locally convex space and E' its dual; suppose that there exists a countable everywhere dense set in E' for the topology σ(E', E). Show that the topology of E (resp. σ(E, E')) is finer than a metrizable locally convex topology. b) Let (x_n) be a sequence of points of E such that every sequence extracted from (x_n) has a limit point for the initial topology (resp. the topology σ(E, E')). Show that there exists a sequence extracted from (x_n) which converges in E for the initial topology (resp. the topology σ(E, E')). (Let (a'_n) be an everywhere dense sequence in E' for σ(E', E); extract a sequence (y_n) from (x_n) such that ⟨y_n, a'_p⟩) tends to a limit for every index p, and show that the sequence (y_n) has only one limit point for the initial topology (resp. for σ(E, E')).) c) For a subset A of E to be relatively compact for the initial topology (resp. for σ(E, E')), it is necessary and sufficient that from every sequence (x_n) of points of A, we can extract a sequence (x_{nk}) which converges to a point of E for the initial topology (resp. for σ(E, E')) (use exerc. 2, a)).