Polar set ($B^{\circ }$)

For a real dual pair $(E,F)$ and $B\subseteq E$, the polar set isThe same absolute-value definition applies to a complex pairing. It collects the linear measurements uniformly bounded by one on $B$. Reversing the roles of $E,F$ defines the bipolar.

A polar is convex, balanced and closed in the topology of pointwise convergence on $E$, since it is an intersection of inverse images of closed scalar unit disks under linear evaluations. Inclusion reverses: $B\subseteq C$ implies $C^{\circ }\subseteq B^{\circ }$.

In $E={\mathbb{R}}^2$ with the usual dot pairing, the polar of $[-1,1{]}^2$ is $\{(a,b):|a|+|b|\le 1\}$, because the maximum of $|ax+by|$ on that square is $|a|+|b|$. In a normed space paired with its continuous dual, the polar of the primal closed unit ball is exactly the dual closed unit ball by the definition of the operator norm.