# 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 $\displaystyle{ (X,Y,\langle , \rangle) }$ the weak topology $\displaystyle{ \sigma(X,Y) }$ is the weakest polar topology on $\displaystyle{ X }$ so that

$\displaystyle{ (X,\sigma(X,Y))' \simeq Y }$.

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

The weak topology is constructed as follows:

For every $\displaystyle{ y }$ in $\displaystyle{ Y }$ on $\displaystyle{ X }$ we define a seminorm on $\displaystyle{ X }$

$\displaystyle{ p_y:X \to \mathbb{R} }$

with

$\displaystyle{ p_y(x) := \vert \langle x , y \rangle \vert \qquad x \in X }$

This family of seminorms defines a locally convex topology on $\displaystyle{ X }$.

## Examples

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