Theorem 1

— A locally convex Hausdorff space is semi-reflexive if and only if every bounded subset of $E$ is relatively compact for the weakened topology $\sigma(E, E')$. If $E$ is semi-reflexive, the strong dual $E'_b$ of $E$ is barrelled.

The second assertion follows from prop. 3 (IV, p. 15), and the identity between bounded subsets for the initial topology and for the weakened topology of $E$ (III, p. 27, cor. 3).

To say that $E$ is semi-reflexive means that the topology on $E'_b$ is compatible with the duality between $E$ and $E'$, in other words, by Mackey's theorem (IV, p. 2, th. 1) that the topology on $E'_b$ is coarser than $\tau(E', E)$ (and in fact is identical with it); by definition (IV, p. 2), this means that every closed, convex and bounded subset of $E$ is compact for $\sigma(E, E')$, and this is equivalent to saying that every bounded subset of E is relatively compact for $\sigma(E, E')$, because the closed convex envelope of a bounded subset of E is bounded (III, p. 3, prop. 1).