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.

Definitions

Let be a dual pair of vector spaces and over the (same) field of real or complex numbers.

A set is said to be bounded in with respect to , if for each element the set of values is bounded in :

This condition is equivalent to the requirement that the polar of the set in

is an absorbent set in , i.e.

Let now be a family of bounded sets in (with respect to ) with the following properties:

Then the seminorms of the form

define a Hausdorff locally convex topology on which is called the polar topology[1] on generated by the family of sets . The sets

form a local base of this topology. A net of elements tends to an element in this topology if and only if

Because of this the polar topology is often called the topology of uniform convergence on the sets of . The semi norm is the gauge of the polar set .

Examples

See also

Notes

References

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