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.

Under the weak topology the bounded sets coincide with the relatively compact sets which leads to the important Bourbaki–Alaoglu theorem.

Definition

Given a dual pair [math]\displaystyle{ (X,Y,\langle , \rangle) }[/math] the weak topology [math]\displaystyle{ \sigma(X,Y) }[/math] is the weakest polar topology on [math]\displaystyle{ X }[/math] so that

[math]\displaystyle{ (X,\sigma(X,Y))' \simeq Y }[/math].

That is the continuous dual of [math]\displaystyle{ (X,\sigma(X,Y)) }[/math] is equal to [math]\displaystyle{ Y }[/math] up to isomorphism.

The weak topology is constructed as follows:

For every [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y }[/math] on [math]\displaystyle{ X }[/math] we define a seminorm on [math]\displaystyle{ X }[/math]

[math]\displaystyle{ p_y:X \to \mathbb{R} }[/math]

with

[math]\displaystyle{ p_y(x) := \vert \langle x , y \rangle \vert \qquad x \in X }[/math]

This family of seminorms defines a locally convex topology on [math]\displaystyle{ X }[/math].

Examples

• Given a normed vector space [math]\displaystyle{ X }[/math] and its continuous dual [math]\displaystyle{ X' }[/math], [math]\displaystyle{ \sigma(X, X') }[/math] is called the weak topology on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \sigma(X', X) }[/math] the weak* topology on [math]\displaystyle{ X' }[/math]