# 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 the **weak topology** is the weakest polar topology on so that

- .

That is the continuous dual of is equal to up to isomorphism.

The weak topology is constructed as follows:

For every in on we define a semi norm on

with

This family of semi norms defines a locally convex topology on .

## Examples

- Given a normed vector space and its continuous dual , is called the weak topology on and the weak* topology on

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