Weak topology (polar topology)

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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 precompact sets which leads to the important Bourbaki-Alaoglu theorem.

[edit] Definition

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

(X,\sigma(X,Y))' \simeq Y.

That is the continuous dual of (X,σ(X,Y)) is equal to Y up to isomorphism.

The weak topology is constructed as follows:

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

p_y:X \to \mathbb{R}

with

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

This family of semi norms defines a locally convex topology on X.

[edit] Examples