Proposition 3

— Let $E$ be a locally convex Hausdorff space. In order that the strong dual $E'_b$ of $E$ be barrelled, it is necessary and sufficient that every subset of $E''$ which is bounded for $\sigma(E'', E')$, is contained in the closure, for $\sigma(E'', E')$, of a bounded subset of $E$.

The equicontinuous subsets of $E''$ are the subsets contained in the bipolar (for the duality between $E''$ and $E'$) of a bounded subset of the subspace $E$ of $E''$. It is now enough to apply the theorem of bipolars (II, p. 45, cor. 3) and the definition of a barrelled space (III, p. 24).