In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS)
equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of
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,
Throughout, all vector spaces will be assumed to be over the field
be a dual pair of vector spaces over the field
This is equivalent to the usual notion of bounded subsets when
which is a Hausdorff locally convex topology.
is understood, is defined as the locally convex topology on
The definition of the strong dual topology now proceeds as in the case of a TVS.
is a TVS whose continuous dual space separates point on
is part of a canonical dual system
is a locally convex space, the strong topology on the (continuous) dual space
(that is, on the space of all continuous linear functionals
and it coincides with the topology of uniform convergence on bounded sets in
runs over the family of all bounded sets in
is a topological vector space (TVS) over the field
be any fundamental system of bounded sets of
forms a fundamental system of bounded sets of
A basis of closed neighborhoods of the origin in
This is a locally convex topology that is given by the set of seminorms on
is identical to the strong dual topology.
endowed with the strong dual topology
is usually assumed to be endowed with the strong dual topology induced on it by
in which case it is called the strong bidual of
is endowed with the strong dual topology
and with the Mackey topology on generated by the pairing
with the strong topology coincides with the Banach dual space
with the topology induced by the operator norm.
is identical to the topology induced by the norm on