Exercise 2, § 3

An infra-barrelled space is semi-barrelled. A locally convex space is said to be a (DF) space if it is semi-barrelled and if the canonical bornology (III, p. 3, def. 5) has a countable base. Every normed space and every strict inductive limit of a sequence of normed spaces (II, p. 33) is a (DF) space. Every strong dual of a Fréchet space is a (DF) space. a) The strong dual of a (DF) space is a Fréchet space. b) Let $E$ be a (DF) space and let $(A_n)$ be an increasing sequence of bounded, convex, balanced and closed subsets of $E$ such that every bounded subset of $E$ is absorbed by one of the $A_n$.

Let U be the union of the $A_n$; show that the closure $\overline{U}$ of U in E is precisely the set of all $x \in E$ such that $\lambda x \in U$ for $0 \leq \lambda < 1$. (If $x \notin \lambda U$ for some $\lambda > 1$, then for every $n$, there exists a linear form $x'_n \in E'$ such that $x'_n \in A_n^\circ$ and $\langle x, x'_n \rangle = \lambda$, and the sequence $(x'_n)$ is equicontinuous, hence has a weak limit point.) c) Show that if a (DF) space is barrelled, it is also bornological ($cf.$ III, p. 44, exerc. 13, b)). Give examples of (DF) spaces which are not ultrabornological, but are bornological and barrelled (III, p. 46, exerc. 22) and also of (DF) spaces which are not barrelled but are bornological.