Strong topology (polar topology)

In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology.

Definition

Given a dual pair <math>(X,Y,\langle , \rangle)<math> the strong topology <math>\beta(Y, X)<math> on <math>Y<math> is the polar topology defined by using the family of all sets in <math>X<math> where the polar set in <math>Y<math> is absorbend.

Examples

• Given a normed vector space <math>X<math> and its continuous dual <math>X'<math> then <math>\beta(X', X)<math>-topology on <math>X'<math> is identical to the topology induced operator norm. Conversely <math>\beta(X, X')<math>-topology on <math>X<math> is identical to the topology induced by the norm.

Properties

• In barrelled spaces the strong topology is identical to the Mackey topology.