In functional analysis and related areas of mathematics a polar topology, topology of ${\displaystyle {\mathcal {G}}}$-convergence or topology of uniform convergence on the sets of ${\displaystyle {\mathcal {G}}}$ is a method to define locally convex topologies on the vector spaces of a pairing.

A pairing is a triple ${\displaystyle (X,Y,b)}$ consisting of two vector spaces over a field ${\displaystyle \mathbb {K} }$ (either the real numbers or complex numbers) and a bilinear map ${\displaystyle b:X\times Y\to \mathbb {K} .}$ A dual pair or dual system is a pairing ${\displaystyle (X,Y,b)}$ satisfying the following two separation axioms:

The polar or absolute polar of a subset ${\displaystyle A\subseteq X}$ is the set

Dually, the polar or absolute polar of a subset ${\displaystyle B\subseteq Y}$ is denoted by ${\displaystyle B^{\circ },}$ and defined by

In this case, the absolute polar of a subset ${\displaystyle B\subseteq Y}$ is also called the prepolar of ${\displaystyle B}$ and may be denoted by ${\displaystyle {}^{\circ }B.}$

The polar is a convex balanced set containing the origin.

If ${\displaystyle A\subseteq X}$ then the bipolar of ${\displaystyle A,}$ denoted by ${\displaystyle A^{\circ \circ },}$ is defined by ${\displaystyle A^{\circ \circ }={}^{\circ }(A^{\circ }).}$ Similarly, if ${\displaystyle B\subseteq Y}$ then the bipolar of ${\displaystyle B}$ is defined to be ${\displaystyle B^{\circ \circ }=\left({}^{\circ }B\right)^{\circ }.}$

Suppose that ${\displaystyle (X,Y,b)}$ is a pairing of vector spaces over ${\displaystyle \mathbb {K} .}$