Weak topology (polar topology)

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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.


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 seminorm on


This family of seminorms defines a locally convex topology on .