Exercise 3, § 3

Let E be a locally convex metrizable space, and $E'_b$ its strong dual. a) Show that every convex balanced subset $V'$ of $E'_b$ which absorbs the strongly bounded subsets of $E'_b$ contains a barrel (for the strong topology) which absorbs the strongly bounded subsets of $E'_b$. (Let $(K'_n)$ be a countable base of the canonical bornology of $E'_b$ and let $\lambda_n$ be such that $\lambda_n K'_n \subset \frac{1}{2}V'$; apply exerc. 2, b) to the sequence $A'_n$, where $A'_n$ is the convex envelope of the union of the $\lambda_j K'_j$ for $j \leq n$.) b) Deduce from a) that the following properties are equivalent : $\alpha)$ E is distinguished (IV, p. 52, exerc. 4). $\beta)$ $E'_b$ is infrabarrelled (III, p. 44, exerc. 7). $\gamma)$ $E'_b$ is bornological. $\delta)$ $E'_b$ is barrelled. $\varepsilon)$ $E'_b$ is ultrabornological (III, p. 45, exerc. 19). c) Show that if $E'_b$ is reflexive, then $\hat{E} = E''$ (which is obviously reflexive) ($cf.$ IV, p. 52, exerc. 4 and p. 53, exerc. 11).