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.
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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
That is the continuous dual of
The weak topology is constructed as follows:
For every
with
This family of semi norms defines a locally convex topology on