Semi-reflexive space

In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable. Suppose that X is a topological vector space (TVS) over the field (which is either the real or complex numbers) whose continuous dual space, , separates points on X (i.e. for any there exists some such that ). Let and both denote the strong dual of X, which is the vector space of continuous linear functionals on X endowed with the topology of uniform convergence on bounded subsets of X; this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified). If X is a normed space, then the strong dual of X is the continuous dual space with its usual norm topology. The bidual of X, denoted by , is the strong dual of ; that is, it is the space . For any let be defined by , where is called the evaluation map at x; since is necessarily continuous, it follows that . Since separates points on X, the map defined by is injective where this map is called the evaluation map or the canonical map. This map was introduced by Hans Hahn in 1927. We call X semireflexive if is bijective (or equivalently, surjective) and we call X reflexive if in addition is an isomorphism of TVSs. If X is a normed space then J is a TVS-embedding as well as an isometry onto its range; furthermore, by Goldstine's theorem (proved in 1938), the range of J is a dense subset of the bidual . A normable space is reflexive if and only if it is semi-reflexive. A Banach space is reflexive if and only if its closed unit ball is -compact.