Weak topology: Information from Answers.com

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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

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 $X$ .

Given a normed vector space $X$ and its continuous dual $X'$ , $σ(X, X')$ is called the weak topology on $X$ and $σ(X', X)$ the weak* topology on $X'$

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