Wikipedia / Reflexive space
Wikipedia / Reflexive space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space into its bidual (which is the strong dual) 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
is a topological vector space (which is either the real or complex numbers) whose continuous dual space separates points) denote the strong dual endowed with the topology of uniform convergence on bounded subsets; this topology is also called the strong dual topology with its usual norm topology. The bidual with its usual norm topology.
Definitions of the evaluation map and reflexive spaces
is a linear map called the is injective where this map is called the evaluation map or the canonical map semi-reflexive is bijective (or equivalently, surjective reflexive 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
is a normed vector space (the real numbers or the complex numbers Consider its dual normed space that consists of all continuous linear functionals and is equipped with the dual norm
is a normed space (a Banach space is called bidual space called evaluation map, that is linear. It follows from the Hahn–Banach theorem
is called reflexive if it satisfies the following equivalent conditions: is surjective, is an isometric isomorphism of normed spaces, is an isomorphism of normed spaces.
Remark
James' space is an example of a non-reflexive space which is linearly isometric to its bidual has codimension one in its bidual. is called quasi-reflexive
Examples
from the definition is bijective, by the rank–nullity theorem.
The Banach space c_0 of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that \ell^1\ell^{\infty}
All Hilbert spaces are reflexive, as are the Lp spaces More generally: all uniformly convex Banach spaces are reflexive according to the Milman–Pettis theorem is not reflexive.
of operators in the Schatten class (the trace class
Properties
Since every finite-dimensional normed space is a reflexive Banach space, only infinite-dimensional spaces can be non-reflexive.
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.
be a Banach space. The following are equivalent. is reflexive. is reflexive. is compact in the weak topology. (This is known as Kakutani's Theorem.) has a weakly convergent subsequence. The statement of Riesz's lemma holds when the real number
on the unit sphere away from the subspace.
is endowed with the usual Euclidean norm plane satisfies the conclusion.
Since norm-closed convex subsets in a Banach space are weakly closed, the intersection is non-empty. As a consequence, every continuous convex function
is a closed non-empty convex is uniformly convex.
A reflexive Banach space is separable
Super-reflexive space
whose two dimensional subspaces are isometric satisfies the parallelogram law, hence is super-reflexive.
is finitely representable such that the multiplicative Banach–Mazur distance
The Lp space
is super-reflexive The notion of ultraproduct of a family of Banach spaces 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 that can be organized in successive levels, starting with level the root that are the children In addition to the tree structure, it is required here that each vector that is an internal vertex
the tree is said to be -separated
Theorem.
Uniformly convex spaces are super-reflexive. be uniformly convex, with modulus of convexity By the properties
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
Reflexive locally convex spaces
The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.
(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 which is called the strong bidual space This induces a map called the evaluation map is locally convex, from the Hahn–Banach theorem). But it can be non-surjective and/or discontinuous.
is called
semi-reflexive is surjective (hence bijective),
reflexive is an isomorphism of topological vector spaces).
Semireflexive spaces
Characterizations
is a Hausdorff locally convex space then the following are equivalent:
is semireflexive;
is weakly compact).
has the weak topology;
is barreled;
is quasi-complete.
Characterizations of reflexive spaces
is a Hausdorff locally convex space then the following are equivalent:
is reflexive;
is semireflexive and infrabarreled;
is semireflexive and barreled;
is barreled is weakly compact).
is semireflexive and quasibarrelled.
is a normed space then the following are equivalent:
is reflexive;
is reflexive.
has nonempty intersection.
Sufficient conditions
Normed spaces
A normed space that is semireflexive is a reflexive Banach space. A closed vector subspace of a reflexive Banach space is reflexive.
are reflexive then they all are. This is why reflexivity is referred to as a .
Topological vector spaces
If a barreled locally convex Hausdorff space is semireflexive then it 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).
Properties
A locally convex Hausdorff reflexive space is barrelled
Examples
Every finite-dimensional Hausdorff topological vector space is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space. and the following conditions become equivalent: is an isomorphism of normed spaces), is an isomorphism of topological vector spaces), is surjective). 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:
the Schwartz space
Counter-examples
There exists a non-reflexive locally convex TVS whose strong dual is reflexive.
Other types of reflexivity
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 is called polar reflexive is an isomorphism of topological vector spaces. (and the stereotype second dual space in the same sense).
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.
– the spaces defined by the corresponding reflexivity condition are called , 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.
See also
A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance is the concept of Grothendieck space.
References
Notes
Citations
General references
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