Exercise 18, § 3

a) Let $E$ be a locally convex Hausdorff space, and let $(A_n)$ be an increasing sequence of convex, balanced, compact sets for the weakened topology $\sigma(E, E')$, such that the union of the sets $A_n$ is $E$, and that for every integer $n$ and every $\lambda > 0$, there exists $m$ such that $\lambda A_n \subset A_m$. Show that $(A_n)$ is a base for the bornology consisting of convex and relatively compact sets for $\sigma(E, E')$. (Observe that on $E'$ the topology of uniform convergence on the $A_n$ is $\tau(E', E)$.) b) If $E$ is barrelled and if there exists a sequence $(A_n)$ having the properties mentioned in $a$, then $E$ is a reflexive (DF) space. (Observe that $E'$, endowed with $\tau(E', E)$ is metrizable and that the topology of $E$ is $\beta(E, E')$.)