Corollary 1

— Suppose that the duality between F and G is separating in F. For a vector subspace M of F the topology $\sigma(G/M^\circ, M)$ is identical with quotient topology of $\sigma(G, F)$ by $M^\circ$, if and only if M is closed for the topology $\sigma(F, G)$.

This follows from prop. 7 putting $N = M^\circ$, and recalling that $M^{\circ \circ}$ is the closure of M for $\sigma(F, G)$ (II, p. 45, cor. 2).