In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual.
Suppose that
is a locally convex space and let
denote the strong dual of
(that is, the continuous dual space of
endowed with the strong dual topology).
denote the continuous dual space of
denote the strong dual of
denote
endowed with the weak-* topology induced by
where this topology is denoted by
(that is, the topology of pointwise convergence on
-bounded if it is a bounded subset of
and we call the closure of
A Hausdorff locally convex space
is called a distinguished space if it satisfies any of the following equivalent conditions: If in addition
is a metrizable locally convex topological vector space then this list may be extended to include: All normed spaces and semi-reflexive spaces are distinguished spaces.
[2] LF spaces are distinguished spaces.
The strong dual space
of a Fréchet space
[3] Every locally convex distinguished space is an H-space.
[2] There exist distinguished Banach spaces spaces that are not semi-reflexive.
[1] The strong dual of a distinguished Banach space is not necessarily separable;
[4] The strong dual space of a distinguished Fréchet space is not necessarily metrizable.
[1] There exists a distinguished semi-reflexive non-reflexive non-quasibarrelled Mackey space
whose strong dual is a non-reflexive Banach space.
[1] There exist H-spaces that are not distinguished spaces.
[1] Fréchet Montel spaces are distinguished spaces.