Theorem 2

— In order that a locally convex Hausdorff space E be reflexive, it is necessary and sufficient that it is barrelled and that every bounded subset of E is relatively compact for the weakened topology $\sigma(E, E')$.

By th. 1 (IV, p. 15), this is the same as saying that E is reflexive if and only if it is semi-reflexive and barrelled.

If E is reflexive, $E'_b$ is reflexive (prop. 4) and consequently E is barrelled (IV, p. 15, th. 1). Conversely, if E is semi-reflexive and barrelled, $c_E$ is a bijection and is bicontinuous by IV, p. 15, prop. 2, hence E is reflexive.