Polar topology

Polar topology

In functional analysis and related areas of mathematics a polar topology, topology of mathcal{A}-convergence or topology of uniform convergence on the sets of mathcal{A} is a method to define locally convex topologies on the vector spaces of a dual pair.

Definition

Given a dual pair (X,Y,langle , angle) and a family mathcal{A} of sets in X such that for all A in mathcal{A} the polar set A^0 is an absorbent subset of Y, the polar topology on Y is defined by a family of semi norms {p_A : A in mathcal{A}}. For each A in mathcal{A} we define :p_A(y):=sup{vert langle x , y angle vert : x in A}.

The semi norm p_A(y) is the gauge of the polar set A^0.

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 sigma(Y,X) on Y. sigma(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 eta(Y,X) on Y

Notes

A polar topology is sometimes called topology of uniform convergence on the sets of mathcal{A} because given a dual pair (X,Y,langle , angle) and a polar topology au on Y defined by the gauges of the polar sets A^0, a sequence y_n in (Y, au) converges to y if and only if for all semi norms p_A:lim_{n o infty} p_A(y_n - y) = lim_{n o infty} sup_{x in A} vert langle y_n - y, x angle vert o 0Or, to put it differently, for all sets A in mathcal{A} :lim_{n o infty} vert langle y_n - y, x angle vert o 0 converges uniformly with respect to x in A.


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