Strong topology (polar topology)

In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology.

Definition

Let be a dual pair of vector spaces over the field of real ( ) or complex ( ) numbers. Let us denote by the system of all subsets bounded by elements of in the following sense:

Then the strong topology on is defined as the locally convex topology on generated by the seminorms of the form

In the special case when is a locally convex space, the strong topology on the (continuous) dual space (i.e. on the space of all continuous linear functionals ) is defined as the strong topology , and it coincides with the topology of uniform convergence on bounded sets in , i.e. with the topology on generated by the seminorms of the form

where runs over the family of all bounded sets in . The space with this topology is called strong dual space of the space and is denoted by .

Examples

  • If is a normed vector space, then its (continuous) dual space with the strong topology coincides with the Banach dual space , i.e. with the space with the topology induced by the operator norm. Conversely -topology on is identical to the topology induced by the norm on .

Properties

  • If is a barrelled space, then its topology coincides with the strong topology on and with the Mackey topology on generated by the pairing .

References

  • Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.
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