Polar set

In functional and convex analysis, and related disciplines of mathematics, the polar set ${\displaystyle A^{\circ }}$ is a special convex set associated to any subset ${\displaystyle A}$ of a vector space ${\displaystyle X,}$ lying in the dual space ${\displaystyle X^{\prime }.}$ The bipolar of a subset is the polar of ${\displaystyle A^{\circ },}$ but lies in ${\displaystyle X}$ (not ${\displaystyle X^{\prime \prime }}$).