Exercise 18, § 4

Let E be a strict inductive limit space of an increasing sequence $(E_n)$ of locally convex metrizable subspaces (II, p. 33), and let F be a vector subspace of E such that every point of E is a limit point of a sequence of points of F. a) If $\overline{E}_n$ is the closure of $E_n$ in E, then E is the strict inductive limit of the sequence $(\overline{E}_n)$. Let $F_n$ be the closure of $F \cap \overline{E}_n$ in E. Show that E is the union of the increasing sequence of sub-spaces $F_n$. b) Suppose E is barrelled. Show that F is bornological. (Let $u$ be a linear mapping from F into a Banach space G which transforms every bounded subset of F into a bounded subset of G. Show that there exists a linear mapping from E into G, whose restriction to F is equal to $u$, and whose restriction to each $F_n$ is continuous. Finally use exerc. 13, b) of III, p. 44.)