Polar topology
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In functional analysis and related areas of mathematics a polar topology, topology of 
-convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a dual pair.
[edit] Definition
Given a dual pair 
and a family of sets in X such that for all A in the polar set A0 is an absorbent subset of Y, the polar topology on Y is defined by a family of semi norms 
. For each A in we define
.
The semi norm pA(y) is the gauge of the polar set A0.
[edit] Examples
- a dual topology is a polar topology (the converse is not necessarily true)
- a locally convex topology is the polar topology defined by the family of equicontinuous sets of the dual space, that is the sets of all continuous linear forms which are equicontinuous
- Using the family of all finite sets in X we get the coarsest polar topology σ(Y,X) on Y. σ(Y,X) is identical to the weak topology.
- Using the family of all sets in X where the polar set is absorbent, we get the finest polar topology β(Y,X) on Y
[edit] Notes
A polar topology is sometimes called topology of uniform convergence on the sets of because given a dual pair and a polar topology τ on Y defined by the gauges of the polar sets A0, a sequence yn in (Y,τ) converges to y if and only if for all semi norms pA
Or, to put it differently, for all sets 
converges uniformly 
.
