Polar topology

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

Definition

Given a dual pair $(X,Y,langle\; ,\; angle)$ and a family $mathcal\{A\}$ of sets in $X$ such that for all $A$ in $mathcal\{A\}$ the polar set $A^0$ is an absorbent subset of $Y$, the polar topology on $Y$ is defined by a family of semi norms $\{p\_A\; :\; A\; in\; mathcal\{A\}\}$. For each $A$ in $mathcal\{A\}$ we define :$p\_A(y):=sup\{vert\; langle\; x\; ,\; y\; angle\; vert\; :\; x\; in\; A\}$.

The semi norm $p\_A(y)$ is the gauge of the polar set $A^0$.

Examples

* a dual topology is a polar topology (the converse is not necessarily true)
* a locally convex topology is the polar topology defined by the family of equicontinuous sets of the dual space, that is the sets of all continuous linear forms which are equicontinuous
* Using the family of all finite sets in $X$ we get the coarsest polar topology $sigma(Y,X)$ on $Y$. $sigma(Y,X)$ is identical to the weak topology.
* Using the family of all sets in $X$ where the polar set is absorbent, we get the finest polar topology on $Y$

Notes

A polar topology is sometimes called topology of uniform convergence on the sets of $mathcal\{A\}$ because given a dual pair $(X,Y,langle\; ,\; angle)$ and a polar topology $au$ on $Y$ defined by the gauges of the polar sets $A^0$, a sequence $y\_n$ in $(Y,\; au)$ converges to $y$ if and only if for all semi norms $p\_A$:$lim\_\{n\; o\; infty\}\; p\_A(y\_n\; -\; y)\; =\; lim\_\{n\; o\; infty\}\; sup\_\{x\; in\; A\}\; vert\; langle\; y\_n\; -\; y,\; x\; angle\; vert\; o\; 0$Or, to put it differently, for all sets $A\; in\; mathcal\{A\}$ :$lim\_\{n\; o\; infty\}\; vert\; langle\; y\_n\; -\; y,\; x\; angle\; vert\; o\; 0$ converges uniformly with respect to $x\; in\; A$.