Corollary 2

— If M is of finite dimension n and the duality is separating in F, then $M^\circ$ is of codimension n in G. If M is closed for $\sigma(F, G)$ and of finite codimension n and if the duality is separating in G, then $M^\circ$ is of dimension n.

For, $G/M^0$ is in separating duality with M; if M is of dimension n, the same is therefore true of $G/M^\circ$ (II, p. 41, example 1). If M is closed, $F/M = F/M^{\circ \circ}$ is in separating duality with $M^\circ$; if F/M is of dimension n, it is therefore the same for $M^\circ$ (II, p. 41, example 1).