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 relatively compact sets which leads to the important Bourbaki–Alaoglu theorem.

Definition[edit]

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

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

That is the continuous dual of (X,\sigma(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.

Examples[edit]