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Concept
Strong dual space
Summary
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space of equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of where this topology is denoted by or The coarsest polar topology is called weak topology.
The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise.
To emphasize that the continuous dual space, has the strong dual topology, or may be written.
Throughout, all vector spaces will be assumed to be over the field of either the real numbers or complex numbers
Dual system
Let be a dual pair of vector spaces over the field of real numbers or complex numbers
For any and any define
Neither nor has a topology so say a subset is said to be if for all
So a subset is called if and only if
This is equivalent to the usual notion of bounded subsets when is given the weak topology induced by which is a Hausdorff locally convex topology.
Let denote the family of all subsets bounded by elements of ; that is, is the set of all subsets such that for every
Then the on also denoted by or simply or if the pairing is understood, is defined as the locally convex topology on generated by the seminorms of the form
The definition of the strong dual topology now proceeds as in the case of a TVS.
Note that if is a TVS whose continuous dual space separates point on then is part of a canonical dual system
where
In the special case when is a locally convex space, the on the (continuous) dual space (that is, 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.
Official source
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