# 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 semi norm 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 semi norms 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 ${\displaystyle X'}$