Concept
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 and denote the strong dual of (that is, the continuous dual space of endowed with the strong dual topology). Let denote the continuous dual space of and let denote the strong dual of Let denote endowed with the weak-* topology induced by where this topology is denoted by (that is, the topology of pointwise convergence on ). We say that a subset of is -bounded if it is a bounded subset of and we call the closure of in the TVS the -closure of . If is a subset of then the polar of is A Hausdorff locally convex space is called a distinguished space if it satisfies any of the following equivalent conditions: If is a -bounded subset of then there exists a bounded subset of whose -closure contains . If is a -bounded subset of then there exists a bounded subset of such that is contained in which is the polar (relative to the duality ) of The strong dual of is a barrelled space. If in addition is a metrizable locally convex topological vector space then this list may be extended to include: (Grothendieck) The strong dual of is a bornological space. All normed spaces and semi-reflexive spaces are distinguished spaces. LF spaces are distinguished spaces. The strong dual space of a Fréchet space is distinguished if and only if is quasibarrelled. Every locally convex distinguished space is an H-space. There exist distinguished Banach spaces spaces that are not semi-reflexive. The strong dual of a distinguished Banach space is not necessarily separable; is such a space. The strong dual space of a distinguished Fréchet space is not necessarily metrizable. There exists a distinguished semi-reflexive non-reflexive -quasibarrelled Mackey space whose strong dual is a non-reflexive Banach space.