Polar topology

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

Definition

Given a dual pair <math>(X,Y,\langle , \rangle)<math> and a family of sets <math>\mathcal{A}<math> in <math>X<math> so that the polar set <math>A^0<math> is an absorbent subset of <math>Y<math> then the polar topology on <math>Y<math> is defined by a family of semi norms <math>\{p_A : A \in \mathcal{A}\}<math>. For each <math>A<math> in <math>\mathcal{A}<math> we define

<math>p_A(y):=\sup\{\vert \langle x , y \rangle \vert : x \in A\}<math>.

The semi norm <math>p_A(y)<math> is the gauge of the polar set <math>A^0<math>.

Examples

• a dual topology is a polar topology (the converse is 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 <math>X<math> we get the coarsest polar topology <math>\sigma(Y,X)<math> on <math>Y<math>. <math>\sigma(Y,X)<math> is identical to the weak topology.
• Using the family of all sets in <math>X<math> where the polar set is absorbend, we get the finest polar topology <math>\beta(Y,X)<math> on <math>Y<math>

Notes

A polar topology is sometimes called topology of uniform convergence on the sets of <math>\mathcal{A}<math> because given a dual pair <math>(X,Y,\langle , \rangle)<math> and a polar topology <math>\tau<math> on <math>Y<math> defined by the gauges of the polar sets <math>A^0<math>, a sequence <math>y_n<math> in <math>(Y, \tau)<math> converges to <math>y<math> if and only if for all semi norms <math>p_A<math>

<math>\lim_{n \to \infty} p_A(y_n – y) = \lim_{n \to \infty} \sup_{x \in A} \vert \langle y_n – y, x \rangle \vert \to 0<math>

Or, to put it differently, for all sets <math>A \in \mathcal{A}<math>

<math>\lim_{n \to \infty} \vert \langle y_n – y, x \rangle \vert \to 0<math> converges uniformly <math>\forall x \in A<math>.