# 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 $(X,Y,\langle ,\rangle )$ 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 seminorm 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 seminorms defines a locally convex topology on $X$ .

## Examples

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