Weak topology (polar topology)

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]