Proposition 4

Let E be a locally convex space. The Mackey topology on E is identical with the initial topology in each of the following cases :

a) E is barrelled; b) E is bornological; c) E is metrizable.

We note first that the Mackey topology on E is identical with the initial topology if and only if every convex subset of $E'$ which is compact for $\sigma(E', E)$, is equi-continuous. This is certainly the case if E is barrelled (III, p. 24, corollary).

Suppose E is bornological; let V be a convex and balanced neighbourhood of 0 in E for the topology $\tau(E, E')$. Let B be a subset of E, bounded for the initial topology. Since B is bounded for the Mackey topology, V absorbs B, and since E is bornological, V is a neighbourhood of 0 for the initial topology.

In case c), the space E is bornological (III, p. 12, prop. 2).