Polar topology

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

Let be a dual pair of vector spaces and over the field , either the 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:

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:

  • each point of belongs to some set
  • each two sets and are contained in some set :
  • is closed under the operation of multiplication by scalars:

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[edit]

  • if is the family of all bounded sets in then the polar topology on coincides with the strong topology,
  • if is the family of all finite sets in then the polar topology on coincides with the weak topology,
  • the topology of an arbitrary locally convex space can be described as the polar topology defined on by the family of all equicontinuous sets in the dual space .[2]

See also[edit]

Notes[edit]

  1. ^ A.P.Robertson, W.Robertson (1964, III.2)
  2. ^ In other words, iff and there is a neighbourhood of zero such that

References[edit]

  • Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press. 
  • Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.