Bipolar theorem

Remarks

The set described is the smallest $\sigma(X, Y)$-closed, convex and balanced set containing $A$, the closed absolutely convex hull of $A$. The bipolar contains $A$ and is convex, balanced and $\sigma(X, Y)$-closed, so it contains this hull. For the reverse inclusion, take $x_0$ outside the hull. The continuous dual of $(X, \sigma(X, Y))$ is $Y$, so the Hahn–Banach separation theorem gives $y \in Y$ with $\abs{\langle a, y \rangle} \le 1$ on the hull and $\abs{\langle x_0, y \rangle} > 1$. Hence $x_0 \notin A^{\circ\circ}$. Example: take a normed space $E$ with the pair $(E, E')$, and let $A$ be the open unit ball of $E$. Then $A^\circ$ is the closed unit ball of $E'$, and $A^{\circ\circ}$ is the closed unit ball of $E$, the weak closure of the open ball. Over $\R$ with one-sided polars $\set{y : \langle x, y \rangle \le 1}$, the bipolar is instead the closed convex hull of $A \cup \set{0}$.

Going deeper

For a student

The bipolar theorem is the convex-geometric form of the Hahn–Banach theorem for a dual pair. It shows that every closed absolutely convex set is an intersection of "slabs" $\set{x : \abs{\langle x, y \rangle} \le 1}$. It is the main tool in the proof of the Mackey–Arens theorem and in describing equicontinuous sets and bounded sets through polars. Combined with the Banach–Alaoglu theorem, it describes the closed absolutely convex neighbourhoods of $0$. Each of them is the polar of its own polar, which is a weak-* compact absolutely convex set of the dual.