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.
Given a dual pair the weak topology is the weakest polar topology on so that
The weak topology is constructed as follows:
For every in on we define a semi norm on
This family of semi norms defines a locally convex topology on .
- Given a normed vector space and its continuous dual , is called the weak topology on and the weak* topology on