Exercise 18, § 3

a) Let E be a separated locally convex 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 $A_n$ is E, and such that for every integer $n$ and every $\lambda > 0$, there exists $m$ such that $\lambda A_n \subset A_m$. Prove that $(A_n)$ is a basis for the bornology formed by the convex and relatively compact sets for $\sigma(E, E')$. (Observe that on $E'$ the topology of uniform convergence in the $A_n$ is $\tau(E', E)$.) b) If E is barrelled and if there exists a sequence $(A_n)$ having the properties of $a$), E is a reflexive (DF) space. (Observe that $E'$, endowed with $\tau(E', E)$ is metrizable and that the topology of E is $\beta(E, E')$.)