Weak topology (polar topology)

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

Definition

Given a dual pair <math>(X,Y,\langle , \rangle)<math> the weak topology <math>\sigma(X,Y)<math> is the weakest polar topology on <math>X<math> so that

<math>(X,\sigma(X,Y))' \simeq Y<math>.

That is the continuous dual of <math>(X,\sigma(X,Y))<math> is equal to <math>Y<math> up to isomorphism.

The weak topology is constructed as follows:

For every <math>y<math> in <math>Y<math> on <math>X<math> we define a semi norm on <math>X<math>

<math>p_y:X \to \mathbb{R}<math>

with

<math>p_y(x) := \vert \langle x , y \rangle \vert \qquad x \in X <math>

This familiy of semi norms defines a locally convex topology on <math>X<math>.

Examples

• Given a normed vector space <math>X<math> and its continuous dual <math>X'<math>, <math>\sigma(X, X')<math> is called the weak topology on <math>X<math> and <math>\sigma(X', X)<math> the weak* topology on <math>X'<math>