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