Exercise 3, § 3

Let $E$ be a locally convex metrizable space, $E'_b$ its strong dual. a) Prove that every convex balanced subset $V'$ of $E'_b$ absorbing the strongly bounded subsets of $E'_b$ contains a barrel (for the strong topology) absorbing 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 exercise 2, b) to the sequence of $A'_n$, where $A'_n$ is the convex hull of the union of the $\lambda_j K'_j$ for $j \leq n$.) b) Deduce from a) that the following properties are equivalent: α) E is distinguished (IV, p. 52, exercise.