In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from into its bidual (which is the strong dual of the strong dual of ) is a homeomorphism (or equivalently, a TVS isomorphism). A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this (always linear) evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces.
In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.
Definition
Definition of the bidual
Suppose that is a topological vector space (TVS) over the field (which is either the real or complex numbers) whose continuous dual space, separates points on (that is, for any there exists some such that ). Let (some texts write ) denote the strong dual of which is the vector space of continuous linear functionals on endowed with the topology of uniform convergence on bounded subsets of ; 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 is a normed space, then the strong dual of is the continuous dual space with its usual norm topology. The bidual of denoted by is the strong dual of ; that is, it is the space If is a normed space, then is the continuous dual space of the Banach space with its usual norm topology.
Definitions of the evaluation map and reflexive spaces
For any let be defined by where is a linear map called the evaluation map at ; since is necessarily continuous, it follows that Since separates points on the linear map defined by is injective where this map is called the evaluation map or the canonical map. Call semi-reflexive if is bijective (or equivalently, surjective) and we call reflexive if in addition is an isomorphism of TVSs. A normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.
Reflexive Banach spaces
Suppose is a normed vector space over the number field or (the real numbers or the complex numbers), with a norm Consider its dual normed space that consists of all continuous linear functionals and is equipped with the dual norm defined by
The dual is a normed space (a Banach space to be precise), and its dual normed space is called bidual space for The bidual consists of all continuous linear functionals and is equipped with the norm dual to Each vector generates a scalar function by the formula:
and is a continuous linear functional on that is, One obtains in this way a map
called evaluation map, that is linear. It follows from the Hahn–Banach theorem that is injective and preserves norms:
that is, maps isometrically onto its image in Furthermore, the image is closed in but it need not be equal to
A normed space is called reflexive if it satisfies the following equivalent conditions: the evaluation map is surjective, the evaluation map is an isometric isomorphism of normed spaces, the evaluation map is an isomorphism of normed spaces. A reflexive space is a Banach space, since is then isometric to the Banach space
Remark
A Banach space is reflexive if it is linearly isometric to its bidual under this canonical embedding James' space is an example of a non-reflexive space which is linearly isometric to its bidual. Furthermore, the image of James' space under the canonical embedding has codimension one in its bidual.
A Banach space is called quasi-reflexive (of order ) if the quotient has finite dimension
Examples
Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection from the definition is bijective, by the rank–nullity theorem. The Banach space of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that and are not reflexive, because is isomorphic to the dual of and is isomorphic to the dual of All Hilbert spaces are reflexive, as are the Lp spaces for More generally: all uniformly convex Banach spaces are reflexive according to the Milman–Pettis theorem. The and spaces are not reflexive (unless they are finite dimensional, which happens for example when is a measure on a finite set). Likewise, the Banach space of continuous functions on is not reflexive. The spaces of operators in the Schatten class on a Hilbert space are uniformly convex, hence reflexive, when When the dimension of is infinite, then (the trace class) is not reflexive, because it contains a subspace isomorphic to and (the bounded linear operators on ) is not reflexive, because it contains a subspace isomorphic to In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of
Properties
Since every finite-dimensional normed space is a reflexive Banach space, only infinite-dimensional spaces can be non-reflexive.
If a Banach space is isomorphic to a reflexive Banach space then is reflexive.Proposition 1.11.8 in .
Every closed linear subspace of a reflexive space is reflexive. The continuous dual of a reflexive space is reflexive. Every quotient of a reflexive space by a closed subspace is reflexive..
Let be a Banach space. The following are equivalent. The space is reflexive. The continuous dual of is reflexive.Corollary 1.11.17, p. 104 in . The closed unit ball of is compact in the weak topology. (This is known as Kakutani's Theorem.) Every bounded sequence in has a weakly convergent subsequence.Since weak compactness and weak sequential compactness coincide by the Eberlein–Šmulian theorem. The statement of Riesz's lemma holds when the real numberThe statement of Riesz's lemma involves only one real number, which is denoted by in the article on Riesz's lemma. The lemma always holds for all real But for a Banach space, the lemma holds for all if and only if the space is reflexive. is exactly Explicitly, for every closed proper vector subspace of there exists some vector of unit norm such that for all Using to denote the distance between the vector and the set this can be restated in simpler language as: is reflexive if and only if for every closed proper vector subspace there is some vector on the unit sphere of that is always at least a distance of away from the subspace. For example, if the reflexive Banach space is endowed with the usual Euclidean norm and is the plane then the points satisfy the conclusion If is instead the -axis then every point belonging to the unit circle in the plane satisfies the conclusion. Every continuous linear functional on attains its supremum on the closed unit ball in Theorem 1.13.11 in . (James' theorem)
Since norm-closed convex subsets in a Banach space are weakly closed,Theorem 2.5.16 in . it follows from the third property that closed bounded convex subsets of a reflexive space are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of the intersection is non-empty. As a consequence, every continuous convex function on a closed convex subset of such that the set
is non-empty and bounded for some real number attains its minimum value on
The promised geometric property of reflexive Banach spaces is the following: if is a closed non-empty convex subset of the reflexive space then for every there exists a such that minimizes the distance between and points of This follows from the preceding result for convex functions, applied to Note that while the minimal distance between and is uniquely defined by the point is not. The closest point is unique when is uniformly convex.
A reflexive Banach space is separable if and only if its continuous dual is separable. This follows from the fact that for every normed space separability of the continuous dual implies separability of Theorem 1.12.11 and Corollary 1.12.12 in .
Super-reflexive space
Informally, a super-reflexive Banach space has the following property: given an arbitrary Banach space if all finite-dimensional subspaces of have a very similar copy sitting somewhere in then must be reflexive. By this definition, the space itself must be reflexive. As an elementary example, every Banach space whose two dimensional subspaces are isometric to subspaces of satisfies the parallelogram law, hencesee this characterization of Hilbert space among Banach spaces is a Hilbert space, therefore is reflexive. So is super-reflexive.
The formal definition does not use isometries, but almost isometries. A Banach space is finitely representableJames, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896–904. in a Banach space if for every finite-dimensional subspace of and every there is a subspace of such that the multiplicative Banach–Mazur distance between and satisfies
A Banach space finitely representable in is a Hilbert space. Every Banach space is finitely representable in The Lp space is finitely representable in
A Banach space is super-reflexive if all Banach spaces finitely representable in are reflexive, or, in other words, if no non-reflexive space is finitely representable in The notion of ultraproduct of a family of Banach spacesDacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math. 41:315–334. allows for a concise definition: the Banach space is super-reflexive when its ultrapowers are reflexive.
James proved that a space is super-reflexive if and only if its dual is super-reflexive.
Finite trees in Banach spaces
One of James' characterizations of super-reflexivity uses the growth of separated trees.see . The description of a vectorial binary tree begins with a rooted binary tree labeled by vectors: a tree of height in a Banach space is a family of vectors of that can be organized in successive levels, starting with level 0 that consists of a single vector the root of the tree, followed, for by a family of 2 vectors forming level
that are the children of vertices of level In addition to the tree structure, it is required here that each vector that is an internal vertex of the tree be the midpoint between its two children:
Given a positive real number the tree is said to be -separated if for every internal vertex, the two children are -separated in the given space norm:
Theorem. The Banach space is super-reflexive if and only if for every there is a number such that every -separated tree contained in the unit ball of has height less than
Uniformly convex spaces are super-reflexive. Let be uniformly convex, with modulus of convexity and let be a real number in By the properties of the modulus of convexity, a -separated tree of height contained in the unit ball, must have all points of level contained in the ball of radius By induction, it follows that all points of level are contained in the ball of radius
If the height was so large that