Theorem 1 (The bipolar theorem)

Let F, G be two real vector spaces in duality. For every subset M of F the polar set $M^{\circ \circ}$ in F of the polar set $M^\circ$ of M in G is the closed convex envelope (for $\sigma(F, G)$) of $M \cup \{0\}$.

We have seen that we need only consider the case where $M$ is convex and $0 \in M$. Denote the closure, in topology $\sigma(F, G)$ of $M$ by $\overline{M}$, then $\overline{M}$ is a convex set in $F$; prop. 4 of II, p. 44 shows that $M^{\circ\circ} \supseteq \overline{M}$. On the other hand if $a \in F$ does not belong to $\overline{M}$ then there exists a closed hyperplane $H$ in $F$ which separates $a$ strictly from $\overline{M}$ (II, p. 38, prop. 4); since $H$ does not contain 0, there exists $y \in G$ such that $H$ has the equation $\langle x, y \rangle = -1$ (II, p. 43, prop. 3); thus $\langle x, y \rangle > -1$ for all $x \in \overline{M}$ and $\langle a, y \rangle < -1$. This implies that $y \in M^\circ$ and $a \notin M^{\circ\circ}$, and the relation $M^{\circ\circ} = \overline{M}$ follows.