In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from into its bidual (which is the strong dual of the strong dual of ) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily not the canonical evaluation map).
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.
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 and both 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.
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.
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:
A reflexive space is a Banach space, since is then isometric to the Banach space
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
If a Banach space is isomorphic to a reflexive Banach space then 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 be a Banach space. The following are equivalent.
Since norm-closed convex subsets in a Banach space are weakly closed, 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 setit follows from the third property that closed bounded convex subsets of a reflexive space
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
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, hence 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 representable 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 spaces 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.
One of James' characterizations of super-reflexivity uses the growth of separated trees. 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 levelThe description of a vectorial binary tree begins with a rooted binary tree labeled by vectors: a tree of height
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. 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 thanThe Banach space
Uniformly convex spaces are super-reflexive. 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 radiusLet
If the height was so large that
then the two points of the first level could not be -separated, contrary to the assumption. This gives the required bound function of only.
Using the tree-characterization, Enflo proved admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant and some real numberthat 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
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 (of real numbers or complex numbers ). 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 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 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 that is,, This induces a map called the evaluation map:
This map is linear. If is locally convex, from the Hahn–Banach theorem it follows that is injective and open (that is, for each neighbourhood of zero in there is a neighbourhood of zero in such that ). But it can be non-surjective and/or discontinuous.
A locally convex space is called
Theorem — A locally convex space is reflexive if and only if it is semi-reflexive and barreled.
Theorem — If is a Hausdorff locally convex space then the canonical injection from into its bidual is a topological embedding if and only if is infrabarreled.
If is a Hausdorff locally convex space then the following are equivalent:
If is a Hausdorff locally convex space then the following are equivalent:
If is a normed space then the following are equivalent:
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.
A normed space that is semireflexive is a reflexive Banach space.A closed vector subspace of a reflexive Banach space is reflexive.
Let be a Banach space and a closed vector subspace of If two of and are reflexive then they all are. This is why reflexivity is referred to as a three-space property.
If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.
The strong dual of a reflexive space is reflexive.The strong dual of a reflexive space is reflexive. Every Montel space is reflexive. And the strong dual of a Montel space is a Montel space (and thus is reflexive).
A locally convex Hausdorff reflexive space is barrelled. If is a normed space then is an isometry onto a closed subspace of This isometry can be expressed by:
Suppose that is a normed space and is its bidual equipped with the bidual norm. Then the unit ball of is dense in the unit ball of for the weak topology
A stereotype space, or polar reflexive space, is defined as a topological vector space (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space More precisely, a TVS is called polar reflexive or stereotype if the evaluation map into the second dual space
is an isomorphism of topological vector spaces. is defined as the space of continuous linear functionals endowed with the topology of uniform convergence on totally bounded sets in (and the stereotype second dual space is the space dual to in the same sense).Here the stereotype dual space
In contrast to the classical reflexive spaces the class Ste of stereotype spaces is very wide (it contains, in particular, all Fréchet spaces and thus, all Banach spaces), it forms a closed monoidal category, and it admits standard operations (defined inside of Ste) 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 (and totally bounded) subsets in in the definition of dual space by other classes of subsets, for example, by the class of compact subsets in – 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 (2012), whether this class forms a category with properties similar to those of Ste.
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