Strong topology (polar topology)

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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[edit]

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[edit]

  • 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[edit]

  • 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[edit]

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