Reflexive space


In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space such that the canonical evaluation map from X into its bidual is an isomorphism of TVSs.
Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective;
in this case the normed space is necessarily also a Banach space.
Note that in 1951, R. C. James discovered a non-reflexive Banach space that is isometrically isomorphic to its bidual.
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 X is a topological vector space over the field whose continuous dual space,, separates points on X.
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.
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.
If X 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 x ∈ X, let be defined by, where Jx is a linear map called the evaluation map at x;
since is necessarily continuous, it follows that.
Since separates points on X, the linear map defined by is injective where this map is called the evaluation map or the canonical map.
We call X semi-reflexive if is bijective and we call X 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.

Semireflexive spaces

Characterizations

If X is a Hausdorff locally convex space then the following are equivalent:
  1. X is semireflexive;
  2. the weak topology on X had the Heine-Borel property.
  3. If linear form on that continuous when has the strong dual topology, then it is continuous when has the weak topology;
  4. is barreled;
  5. X weak the weak topology is quasi-complete.

    Reflexive spaces

Characterizations

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

  1. is reflexive;
  2. is semireflexive and infrabarreled;
  3. is semireflexive and barreled;
  4. is barreled and the weak topology on had the Heine-Borel property.
  5. is semireflexive and quasibarrelled.
If is a normed space then the following are equivalent:

  1. is reflexive;
  2. the closed unit ball is compact when has the weak topology.
  3. is a Banach space and is reflexive.
Theorem: A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.

Sufficient conditions

Suppose is a normed vector space over the number field or , 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, 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, i.e.,. 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:
i.e., 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:
A reflexive space is a Banach space, since is then isometric to the Banach space.

Remark

A Banach space X is reflexive if it is linearly isometric to its bidual under this canonical embedding J. 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 J has codimension one in its bidual.
A Banach space X is called quasi-reflexive if the quotient has finite dimension d.

Examples

1) 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 J from the definition is bijective, by the rank–nullity theorem.
2) The Banach space c0 of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that ℓ1 and ℓ∞ are not reflexive, because ℓ1 is isomorphic to the dual of c0, and ℓ∞ is isomorphic to the dual of ℓ1.
3) 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 L1 and L∞ spaces are not reflexive. Likewise, the Banach space C of continuous functions on is not reflexive.
4) The spaces Sp of operators in the Schatten class on a Hilbert space H are uniformly convex, hence reflexive, when. When the dimension of H is infinite, then S1 is not reflexive, because it contains a subspace isomorphic to ℓ1, and S∞ = L is not reflexive, because it contains a subspace isomorphic to ℓ∞. In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of H.

Properties

If a Banach space Y is isomorphic to a reflexive Banach space X, then Y is reflexive.
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 X be a Banach space. The following are equivalent.
  1. The space X is reflexive.
  2. The continuous dual of X is reflexive.
  3. The closed unit ball of X is compact in the weak topology.
  4. Every bounded sequence in X has a weakly convergent subsequence.
  5. Every continuous linear functional on X attains its maximum on the closed unit ball in X.
Since norm-closed convex subsets in a Banach space are weakly closed,
it follows from the third property that closed bounded convex subsets of a reflexive space X are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of X, the intersection is non-empty. As a consequence, every continuous convex function f on a closed convex subset C of X, such that the set
is non-empty and bounded for some real number t, attains its minimum value on C.
The promised geometric property of reflexive Banach spaces is the following: if C is a closed non-empty convex subset of the reflexive space X, then for every x in X there exists a c in C such that minimizes the distance between x and points of C. This follows from the preceding result for convex functions, applied to. Note that while the minimal distance between x and C is uniquely defined by x, the point c is not. The closest point c is unique when X 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 Y, separability of the continuous dual implies separability.

Super-reflexive space

Informally, a super-reflexive Banach space X has the following property: given an arbitrary Banach space Y, if all finite-dimensional subspaces of Y have a very similar copy sitting somewhere in X, then Y must be reflexive. By this definition, the space X itself must be reflexive. As an elementary example, every Banach space Y whose two dimensional subspaces are isometric to subspaces of satisfies the parallelogram law, hence
Y is a Hilbert space, therefore Y is reflexive. So ℓ2 is super-reflexive.
The formal definition does not use isometries, but almost isometries. A Banach space Y is finitely representable
in a Banach space X if for every finite-dimensional subspace Y0 of Y and every, there is a subspace X0 of X such that the multiplicative Banach-Mazur distance between X0 and Y0 satisfies
A Banach space finitely representable in ℓ2 is a Hilbert space. Every Banach space is finitely representable in c0. The space Lp space|Lp is finitely representable in ℓp.
A Banach space X is super-reflexive if all Banach spaces Y finitely representable in X are reflexive, or, in other words, if no non-reflexive space Y is finitely representable in X. The notion of ultraproduct of a family of Banach spaces
allows for a concise definition: the Banach space X 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.
The description of a vectorial binary tree begins with a rooted binary tree labeled by vectors: a tree of height n in a Banach space X is a family of vectors of X, that can be organized in successive levels, starting with level 0 that consists of a single vector x∅, the root of the tree, followed, for, by a family of 2k vectors forming level k:
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 t, the tree is said to be t-separated if for every internal vertex, the two children are t-separated in the given space norm:
Theorem.
The Banach space X is super-reflexive if and only if for every, there is a number n such that every t-separated tree contained in the unit ball of X has height less than n.

Uniformly convex spaces are super-reflexive.
Let X be uniformly convex, with modulus of convexity δX and let t be a real number in . By the properties of the modulus of convexity, a t-separated tree of height n, 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 n was so large that
then the two points x1, x−1 of the first level could not be t-separated, contrary to the assumption. This gives the required bound n, function of δX only.
Using the tree-characterization, Enflo proved
that super-reflexive Banach spaces admit an equivalent uniformly convex norm. Trees in a Banach space are a special instance of vector-valued martingales. Adding techniques from scalar martingale theory, Pisier improved Enflo's result by showing
that a super-reflexive space X admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant and some real number ,

Reflexive locally convex spaces

The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.
Let be a topological vector space over a number field . Consider its strong dual space, which consists of all continuous linear functionals and is equipped with the strong topology, i.e., the topology of uniform convergence on bounded subsets in. The space is a topological vector space, so one can consider its strong dual space, which is called the strong bidual space for. It consists of all
continuous linear functionals and is equipped with the strong topology. Each vector generates a map by the following formula:
This is a continuous linear functional on, i.e.,. One obtains a map called evaluation map:
This map is linear. If is locally convex, from the Hahn–Banach theorem it follows that is injective and open. But it can be non-surjective and/or discontinuous.
A locally convex space is called
Theorem. A locally convex Hausdorff space is semi-reflexive if and only if with the -topology has the Heine–Borel property.
Theorem. A locally convex space is reflexive if and only if it is semi-reflexive and barreled.
Theorem. The strong dual of a semireflexive space is barrelled.

Examples

1) Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
2) A normed space is reflexive as a normed space if and only if it is reflexive as a locally convex space. This follows from the fact that for a normed space its dual normed space coincides as a topological vector space with the strong dual space. As a corollary, the evaluation map coincides with the evaluation map, and the following conditions become equivalent:
3) A example of a semi-reflexive space that is not reflexive is obtained as follows: let Y be an infinite dimensional reflexive Banach space, and let X be the topological vector space, that is, the vector space Y equipped with the weak topology. Then the continuous dual of X and are the same set of functionals, and bounded subsets of X are norm-bounded, hence the Banach space is the strong dual of X. Since Y is reflexive, the continuous dual of is equal to the image J of X under the canonical embedding J, but the topology on X is not the strong topology , that is equal to the norm topology of Y.
4) Montel spaces are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:
A stereotype space, or polar reflexive space, is defined as a topological vector space satisfying a similar condition of reflexivity, but with the topology of uniform convergence on totally bounded subsets in the definition of dual space X’. More precisely, a topological vector space is called polar reflexive or stereotype if the evaluation map into the second dual space
is an isomorphism of topological vector spaces. Here the stereotype dual space is defined as the space of continuous linear functionals endowed with the topology of uniform convergence on totally bounded sets in .
In contrast to the classical reflexive spaces the class Ste of stereotype spaces is very wide, it forms a closed monoidal category, and it admits standard operations of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category Ste have applications in duality theory for non-commutative groups.
Similarly, one can replace the class of bounded subsets in X in the definition of dual space X’, by other classes of subsets, for example, by the class of compact subsets in X – the spaces defined by the corresponding reflexivity condition are called reflective, and they form an even wider class than Ste, but it is not clear, whether this class forms a category with properties similar to those of Ste.