Exercise 17, § 3

a) Let $E$ be a locally convex Hausdorff space, such that the bornology consisting of the relatively compact sets of $E$ has a countable base $(A_n)$ (III, p. 1). Show that $(A_n)$ is also a base for the canonical bornology (III, p. 3, def. 5). (Let $C_n$ be the relatively compact set which is the sum of $n$ sets each equal to $A_n$; then $(C_n)$ is also a base for the bornology of relatively compact sets. Argue by reductio ad absurdum, considering a bounded set $B$ which is not contained in any 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.) Then, the space $E$ is semi-reflexive and the closed convex balanced envelope of every compact set in $E$ is compact. b) Suppose that $E$ is infrabarrelled and that for a topology $\mathscr{T}$ compatible with the duality between $E$ and $E'$, there exists a countable base for the bornology of relatively compact subsets of $E$ for $\mathscr{T}$. Show that then $E$ is a reflexive (DF) space (IV, p. 57, exerc. 2). (Use $a$), observing that the closed bounded sets in $E$ are compact for $\mathscr{T}$, and consequently, complete for the initial topology of $E$; this implies that $E$ is barrelled). If $\mathscr{T}$ is the initial topology, then $E$ is a Montel space.