# Polar topology

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

## Definitions

${\displaystyle \forall y\in Y\qquad \sup _{x\in A}|\langle x,y\rangle |<\infty .}$

This condition is equivalent to the requirement that the polar ${\displaystyle A^{\circ }}$ of the set ${\displaystyle A}$ in ${\displaystyle Y}$

${\displaystyle A^{\circ }=\{y\in Y:\quad \sup _{x\in A}|\langle x,y\rangle |\leq 1\}}$

is an absorbent set in ${\displaystyle Y}$, i.e.

${\displaystyle \bigcup _{\lambda \in {\mathbb {F} }}\lambda \cdot A^{\circ }=Y.}$

Let now ${\displaystyle {\mathcal {A}}}$ be a family of bounded sets in ${\displaystyle X}$ (with respect to ${\displaystyle Y}$) with the following properties:

${\displaystyle \forall x\in X\qquad \exists A\in {\mathcal {A}}\qquad x\in A,}$
${\displaystyle \forall A,B\in {\mathcal {A}}\qquad \exists C\in {\mathcal {A}}\qquad A\cup B\subseteq C,}$
${\displaystyle \forall A\in {\mathcal {A}}\qquad \forall \lambda \in {\mathbb {F} }\qquad \lambda \cdot A\in {\mathcal {A}}.}$

Then the seminorms of the form

${\displaystyle \|y\|_{A}=\sup _{x\in A}|\langle x,y\rangle |,\qquad A\in {\mathcal {A}},}$

define a Hausdorff locally convex topology on ${\displaystyle Y}$ which is called the polar topology[1] on ${\displaystyle Y}$ generated by the family of sets ${\displaystyle {\mathcal {A}}}$. The sets

${\displaystyle U_{B}=\{x\in V:\quad \|\varphi \|_{B}<1\},\qquad B\in {\mathcal {B}},}$

form a local base of this topology. A net of elements ${\displaystyle y_{i}\in Y}$ tends to an element ${\displaystyle y\in Y}$ in this topology if and only if

${\displaystyle \forall A\in {\mathcal {A}}\qquad \|y_{i}-y\|_{A}=\sup _{x\in A}|\langle x,y_{i}\rangle -\langle x,y\rangle |{\underset {i\to \infty }{\longrightarrow }}0.}$

Because of this the polar topology is often called the topology of uniform convergence on the sets of ${\displaystyle {\mathcal {A}}}$. The semi norm ${\displaystyle \|y\|_{A}}$ is the gauge of the polar set ${\displaystyle A^{\circ }}$.