Exercise 8, § 2

Let E be a Hausdorff locally convex space which is the strict inductive limit of an increasing sequence $(E_n)$ of closed vector subspaces (II, p. 32, prop. 9). a) Show that, if the strong dual of each of the E_n is complete, then the strong dual of E is complete (III, p. 20, th. 1). b) In order that E be semi-reflexive (resp. reflexive), it is necessary and sufficient that each of the E_n be semi-reflexive (resp. reflexive).