Corollary 3

Let $\mathscr{T}$ be the topology of a locally convex space $E$ and let $E'$ be its dual (II, p. 42, def. 1).

(i) *The closed convex sets in E are the same for the topology $\mathscr{T}$ and for the weak topology $\sigma(E, E')$.

(ii) *For every subset M of E, the polar set $M^{\circ\circ}$ in E of the polar set $M^\circ$ of M in $E'$, is the convex closed envelope of $M \cup \{0\}$ for the topology $\mathscr{T}$.

Clearly, (ii) follows from (i) and th. 1. From the definition of the dual $E'$, it follows from II, p. 43, prop. 3 that the continuous linear forms on E for the topology $\mathscr{T}$ are the same as the continuous linear forms for $\sigma(E, E')$. The closed half-spaces in E are therefore the same for $\mathscr{T}$ and for $\sigma(E, E')$ (II, p. 15, prop. 17) and the assertion (i) follows therefore from II, p. 38, cor. 1.