Exercise 17, § 3

a) Let E be a separated locally convex space, such that the bornology formed by the relatively compact sets of E admits a countable basis (III, p. 1) $(A_n)$. Now prove that $(A_n)$ is also a basis of the canonical bornology (III, p. 3, déf. 5). (Let $C_n$ be the relatively compact set which is the sum of $n$ sets equal to $A_n$; then $(C_n)$ is also a basis of the bornology of relatively compact sets. Reason by contradiction by considering a bounded set B which is contained in none of the $C_n$, and conclude that there exists a sequence $(x_n)$ of points of B such that $x_n / n \notin A_n$; deduce a contradiction.) The space E is then semi-reflexive and the closed convex balanced envelope of every compact set in E is compact. b) Suppose that E is infratonné and that for a topology $\mathscr{T}$ compatible with the duality between E and $E'$, there exists a countable basis of the bornology of the relatively compact subsets of E for $\mathscr{T}$. Now prove that E is a reflexive (DF) space (IV, p. 58, exerc. 2).

(Use $a$), by observing that the closed bounded sets in E are compact for $\mathscr{T}$, and consequently complete for the initial topology of E, which implies that E is tonnelé). If $\mathscr{T}$ is the initial topology, E is a Montel space.