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 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.

Definition and notation

Brief definition

Suppose that X is a topological vector space (TVS) over the field \mathbb{F} (which is either the real or complex numbers) whose continuous dual space, X^{\prime}, separates points on X (i.e. for any x \in X there exists some such that ). Let and both denote the strong dual of X, which is the vector space X^{\prime} 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 X^{\prime} with its usual norm topology. The bidual of X, denoted by, is the strong dual of ; that is, it is the space. For any x \in X, let be defined by, where J_x is called the evaluation map at x; since is necessarily continuous, it follows that. Since X^{\prime} separates points on X, the map defined by J(x) := J_x 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.

Detailed definition

Let X be a topological vector space over a number field \mathbb{F} (of real numbers \R or complex numbers \C). Consider its strong dual space, which consists of all continuous linear functionals and is equipped with the strong topology , that is, the topology of uniform convergence on bounded subsets in X. The space is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space , which is called the strong bidual space for X. It consists of all continuous linear functionals and is equipped with the strong topology. Each vector x\in X generates a map by the following formula: This is a continuous linear functional on, that is,. One obtains a map called the evaluation map or the canonical injection: which is a linear map. If X is locally convex, from the Hahn–Banach theorem it follows that J is injective and open (that is, for each neighbourhood of zero U in X there is a neighbourhood of zero V in such that ). But it can be non-surjective and/or discontinuous. A locally convex space X is called semi-reflexive if the evaluation map is surjective (hence bijective); it is called reflexive if the evaluation map is surjective and continuous, in which case J will be an isomorphism of TVSs).

Characterizations of semi-reflexive spaces

If X is a Hausdorff locally convex space then the following are equivalent:

Sufficient conditions

Every semi-Montel space is semi-reflexive and every Montel space is reflexive.

Properties

If X is a Hausdorff locally convex space then the canonical injection from X into its bidual is a topological embedding if and only if X is infrabarrelled. The strong dual of a semireflexive space is barrelled. Every semi-reflexive space is quasi-complete. Every semi-reflexive normed space is a reflexive Banach space. The strong dual of a semireflexive space is barrelled.

Reflexive spaces

If X is a Hausdorff locally convex space then the following are equivalent: If X is a normed space then the following are equivalent:

Examples

Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive. If X is a dense proper vector subspace of a reflexive Banach space then X is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space. There exists a semi-reflexive countably barrelled space that is not barrelled.

Citations