{"history":[],"article":{"id":"72b8ba09-ca93-4fd9-be9c-8d660893c11b","title":"Reflexive space","slug":"reflexive-space","content":":::summary\n:::\n\n# \nIn the area of mathematics known as [functional analysis](https://bliptext.com/articles/functional-analysis), a **reflexive space** is a [locally convex](https://bliptext.com/articles/locally-convex) [topological vector space](https://bliptext.com/articles/topological-vector-space) for which the canonical evaluation map from X into its [bidual](https://bliptext.com/articles/bidual) (which is the [strong dual](https://bliptext.com/articles/strong-dual) of the [strong dual](https://bliptext.com/articles/strong-dual) of X) is a [homeomorphism](https://bliptext.com/articles/homeomorphism) (or equivalently, a [TVS isomorphism](https://bliptext.com/articles/tvs-isomorphism)).\nA [normed space](https://bliptext.com/articles/normed-space) is reflexive if and only if this canonical evaluation map is [surjective](https://bliptext.com/articles/surjective), in which case this (always linear) evaluation map is an [isometric isomorphism](https://bliptext.com/articles/isometric-isomorphism) and the [normed space](https://bliptext.com/articles/normed-space) is a [Banach space](https://bliptext.com/articles/banach-space).\nThose spaces for which the canonical evaluation map is surjective are called [semi-reflexive](https://bliptext.com/articles/semi-reflexive) spaces.\nIn 1951, [R. C. James](https://bliptext.com/articles/robert-c-james) discovered a Banach space, now known as [James' space](https://bliptext.com/articles/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](https://bliptext.com/articles/isometric-isomorphism) is necessarily the canonical evaluation map).\nSo 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.\nReflexive spaces play an important role in the general theory of [locally convex](https://bliptext.com/articles/locally-convex) TVSs and in the theory of [Banach spaces](https://bliptext.com/articles/banach-space) in particular.\n[Hilbert spaces](https://bliptext.com/articles/hilbert-space) are prominent examples of reflexive Banach spaces.\nReflexive Banach spaces are often characterized by their geometric properties.\n\n# Definition\nSuppose that X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over the field \\mathbb{F} (which is either the real or complex numbers) whose [continuous dual space](https://bliptext.com/articles/continuous-dual-space), X^{\\prime}, **separates points** on X (that is, for any there exists some such that ).\nLet (some texts write ) denote the [strong dual](https://bliptext.com/articles/strong-dual) of X, which is the vector space X^{\\prime} of continuous linear functionals on X endowed with the [topology of uniform convergence](https://bliptext.com/articles/topology-of-uniform-convergence) on [bounded subsets](https://bliptext.com/articles/topology-of-uniform-convergence) of X; this topology is also called the **[strong dual](https://bliptext.com/articles/strong-dual) topology** and it is the \"default\" topology placed on a continuous dual space (unless another topology is specified).\nIf X is a normed space, then the strong dual of X is the continuous dual space X^{\\prime} with its usual norm topology.\nThe **bidual** of X, denoted by is the strong dual of ; that is, it is the space\nIf X is a normed space, then is the continuous dual space of the Banach space with its usual norm topology.\nFor any x \\in X, let be defined by where J_x is a linear map called the **evaluation map at x**;\nsince is necessarily continuous, it follows that\nSince X^{\\prime} separates points on X, the linear map defined by J(x) := J_x is injective where this map is called the **evaluation map** or the **canonical map**.\nCall X **semi-**reflexive**** if is bijective (or equivalently, [surjective](https://bliptext.com/articles/surjective)) and we call X **reflexive** if in addition is an isomorphism of TVSs.\nA [normable](https://bliptext.com/articles/normable) space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.\n\n# Reflexive Banach spaces\nSuppose X is a [normed vector space](https://bliptext.com/articles/normed-vector-space) over the number field or (the [real numbers](https://bliptext.com/articles/real-number) or the [complex numbers](https://bliptext.com/articles/complex-number)), with a norm Consider its [dual normed space](https://bliptext.com/articles/dual-norm) X^{\\prime}, that consists of all [continuous](https://bliptext.com/articles/continuous-function) [linear functionals](https://bliptext.com/articles/linear-functional) and is equipped with the [dual norm](https://bliptext.com/articles/dual-norm) defined by\nThe dual X^{\\prime} is a normed space (a [Banach space](https://bliptext.com/articles/banach-space) to be precise), and its dual normed space is called **bidual space** for X. The bidual consists of all continuous linear functionals and is equipped with the norm dual to Each vector x \\in X generates a scalar function by the formula:\nand J(x) is a continuous linear functional on X^{\\prime}, that is, One obtains in this way a map\ncalled **evaluation map**, that is linear.\nIt follows from the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) that J is injective and preserves norms:\nthat is, J maps X isometrically onto its image J(X) in Furthermore, the image J(X) is closed in but it need not be equal to\nA normed space X is called **reflexive** if it satisfies the following equivalent conditions:\n

the evaluation map is [surjective](https://bliptext.com/articles/bijection-injection-and-surjection),

the evaluation map is an [isometric isomorphism](https://bliptext.com/articles/banach-space) of normed spaces,

the evaluation map is an [isomorphism](https://bliptext.com/articles/banach-space) of normed spaces.

\n\nA reflexive space X is a Banach space, since X is then isometric to the Banach space\n\n## Remark\nA Banach space X is reflexive if it is linearly isometric to its [bidual](https://bliptext.com/articles/dual-space) under this canonical embedding J. [James' space](https://bliptext.com/articles/james-space) is an example of a non-reflexive space which is linearly isometric to its [bidual](https://bliptext.com/articles/dual-space).\nFurthermore, the image of James' space under the canonical embedding J has [codimension](https://bliptext.com/articles/codimension) one in its bidual.\nA Banach space X is called **quasi-reflexive** (of order d) if the quotient has finite dimension d.\n\n## Examples\n\n## Properties\nSince every finite-dimensional normed space is a reflexive [Banach space](https://bliptext.com/articles/banach-space), only infinite-dimensional spaces can be non-reflexive.\nIf a Banach space Y is isomorphic to a reflexive Banach space X then Y is reflexive.\nEvery [closed](https://bliptext.com/articles/closed-set) [linear subspace](https://bliptext.com/articles/linear-subspace) of a reflexive space is reflexive.\nThe continuous dual of a reflexive space is reflexive.\nEvery [quotient](https://bliptext.com/articles/quotient-space-linear-algebra) of a reflexive space by a closed subspace is reflexive.\nLet X be a Banach space.\nThe following are equivalent.\n

The space X is reflexive.
The continuous dual of X is reflexive.\n
The closed unit ball of X is [compact](https://bliptext.com/articles/compact-space) in the [weak topology](https://bliptext.com/articles/weak-topology).\n(This is known as Kakutani's Theorem.)
Every bounded sequence in X has a weakly convergent subsequence.\n
The statement of [Riesz's lemma](https://bliptext.com/articles/riesz-s-lemma) holds when the real number is exactly 1.\nExplicitly, for every closed proper vector subspace Y of X, there exists some vector u \\in X of unit norm \\|u\\| = 1 such that for all y \\in Y.
Every continuous linear functional on X attains its supremum on the closed unit ball in X. ([James' theorem](https://bliptext.com/articles/james-theorem))

\nSince norm-closed [convex subsets](https://bliptext.com/articles/convex-set) in a Banach space are weakly closed,\nit follows from the third property that closed bounded convex subsets of a reflexive space X are weakly compact.\nThus, for every decreasing sequence of non-empty closed bounded convex subsets of X, the intersection is non-empty.\nAs a consequence, every continuous [convex function](https://bliptext.com/articles/convex-function) f on a closed convex subset C of X, such that the set\nis non-empty and bounded for some real number t, attains its minimum value on C.\nThe promised geometric property of reflexive Banach spaces is the following: if C is a closed non-empty [convex](https://bliptext.com/articles/convex-set) subset of the reflexive space X, then for every x \\in X there exists a c \\in C such that \\|x - c\\| minimizes the distance between x and points of C. This follows from the preceding result for [convex](https://bliptext.com/articles/convex-set) functions, applied to Note that while the minimal distance between x and C is uniquely defined by x, the point c is not.\nThe closest point c is unique when X is uniformly convex.\nA reflexive Banach space is [separable](https://bliptext.com/articles/separable-space) if and only if its continuous dual is [separable](https://bliptext.com/articles/separable-space).\nThis follows from the fact that for every normed space Y, separability of the continuous dual Y^{\\prime} implies separability of Y.\n\n## Super-reflexive space\nInformally, 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.\nBy this definition, the space X itself must be reflexive.\nAs an elementary example, every Banach space Y whose two dimensional subspaces are [isometric](https://bliptext.com/articles/isometry) to subspaces of X = \\ell^2 satisfies the [parallelogram law](https://bliptext.com/articles/parallelogram-law), hence\nY is a Hilbert space, therefore Y is reflexive.\nSo \\ell^2 is super-reflexive.\nThe formal definition does not use isometries, but almost isometries.\nA Banach space Y is **finitely representable**\nin a Banach space X if for every finite-dimensional subspace Y_0 of Y and every there is a subspace X_0 of X such that the multiplicative [Banach–Mazur distance](https://bliptext.com/articles/banach-mazur-compactum) between X_0 and Y_0 satisfies\nA Banach space finitely representable in \\ell^2 is a Hilbert space.\nEvery Banach space is finitely representable in c_0.\nThe [Lp space](https://bliptext.com/articles/lp-space) L^p([0, 1]) is finitely representable in \\ell^p.\nA 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](https://bliptext.com/articles/ultraproduct) of a family of Banach spaces\nallows for a concise definition: the Banach space X is super-reflexive when its ultrapowers are reflexive.\nJames proved that a space is super-reflexive if and only if its dual is super-reflexive.\n\n## Finite trees in Banach spaces\nOne of James' characterizations of super-reflexivity uses the growth of separated trees.\nThe description of a vectorial binary tree begins with a [rooted binary tree](https://bliptext.com/articles/[root](https://bliptext.com/articles/tree-graph-theory)ed-binary-tree) labeled by vectors: a tree of [height](https://bliptext.com/articles/tree-graph-theory) n in a Banach space X is a family of 2^{n+1} - 1 vectors of X, that can be organized in successive levels, starting with level 0 that consists of a single vector the [root](https://bliptext.com/articles/tree-graph-theory) of the tree, followed, for by a family of s^k2 vectors forming level k:\nthat are the [children](https://bliptext.com/articles/tree-graph-theory) of vertices of level k - 1.\nIn addition to the [tree structure](https://bliptext.com/articles/tree-graph-theory), it is required here that each vector that is an [internal vertex](https://bliptext.com/articles/tree-graph-theory) of the tree be the midpoint between its two children:\nGiven 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:\n**Theorem.**\nThe Banach space X is super-reflexive if and only if for every there is a number n(t) such that every t-separated tree contained in the unit ball of X has height less than n(t).\n[Uniformly convex spaces](https://bliptext.com/articles/uniformly-convex-space) are super-reflexive.\nLet X be uniformly convex, with [modulus of convexity](https://bliptext.com/articles/modulus-and-characteristic-of-convexity) \\delta_X and let t be a real number in (0, 2]. By the [properties](https://bliptext.com/articles/modulus-and-characteristic-of-convexity) of the [modulus of convexity](https://bliptext.com/articles/modulus-and-characteristic-of-convexity), a t-separated tree of height n, contained in the unit ball, must have all points of level n - 1 contained in the ball of radius By induction, it follows that all points of level n - k are contained in the ball of radius\nIf the height n was so large that\nthen the two points x_1, x_{-1} of the first level could not be t-separated, contrary to the assumption.\nThis gives the required bound n(t), function of \\delta_X(t) only.\nUsing the tree-characterization, [Enflo](https://bliptext.com/articles/per-enflo) proved\nthat super-reflexive Banach spaces admit an equivalent uniformly convex norm.\nTrees in a Banach space are a special instance of vector-valued [martingales](https://bliptext.com/articles/martingale-probability-theory).\nAdding techniques from scalar martingale theory, [Pisier](https://bliptext.com/articles/gilles-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 c > 0 and some real number q \\geq 2,\n\n# Reflexive locally convex spaces\nThe notion of reflexive Banach space can be generalized to [topological vector spaces](https://bliptext.com/articles/topological-vector-space) in the following way.\nLet X be a topological vector space over a number field \\mathbb F (of [real numbers](https://bliptext.com/articles/real-number) \\mathbb R or [complex numbers](https://bliptext.com/articles/complex-number) \\Complex).\nConsider its [strong dual space](https://bliptext.com/articles/strong-topology-polar-topology) which consists of all [continuous](https://bliptext.com/articles/continuous-function) [linear functionals](https://bliptext.com/articles/linear-functional) and is equipped with the [strong topology](https://bliptext.com/articles/strong-topology-polar-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](https://bliptext.com/articles/strong-topology-polar-topology) which is called the **strong bidual space** for X. It consists of all [continuous](https://bliptext.com/articles/continuous-function) [linear functionals](https://bliptext.com/articles/linear-functional) and is equipped with the [strong topology](https://bliptext.com/articles/strong-topology-polar-topology) Each vector x \\in X generates a map by the following formula:\nThis is a continuous linear functional on that is,, This induces a map called the **evaluation map**:\nThis map is linear.\nIf X is locally convex, from the [Hahn–Banach theorem](https://bliptext.com/articles/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 ).\nBut it can be non-surjective and/or discontinuous.\nA locally convex space X is called\n\n## Semireflexive spaces\n\n### Characterizations\nIf X is a Hausdorff locally convex space then the following are equivalent:\n\n## Characterizations of reflexive spaces\nIf X is a Hausdorff locally convex space then the following are equivalent:\nIf X is a normed space then the following are equivalent:\n\n## Sufficient conditions\nA normed space that is semireflexive is a reflexive Banach space.\nA closed vector subspace of a reflexive Banach space is reflexive.\nLet X be a Banach space and M a closed vector subspace of X. If two of X, M, and X / M are reflexive then they all are.\nThis is why reflexivity is referred to as a.\nIf a [barreled](https://bliptext.com/articles/barreled-space) locally convex Hausdorff space is semireflexive then it is reflexive.\nThe strong dual of a reflexive space is reflexive.Every [Montel space](https://bliptext.com/articles/montel-space) is reflexive.\nAnd the strong dual of a [Montel space](https://bliptext.com/articles/montel-space) is a [Montel space](https://bliptext.com/articles/montel-space) (and thus is reflexive).\n\n## Properties\nA locally convex Hausdorff reflexive space is [barrelled](https://bliptext.com/articles/barrelled-space).\nIf X is a normed space then is an isometry onto a closed subspace of This isometry can be expressed by:\nSuppose that X is a normed space and is its bidual equipped with the bidual norm.\nThen the unit ball of X, is dense in the unit ball\nof for the weak topology\n\n## Examples\n

Every finite-dimensional Hausdorff [topological vector space](https://bliptext.com/articles/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.
A normed space X is reflexive as a normed space if and only if it is reflexive as a locally convex space.\nThis follows from the fact that for a normed space X its dual normed space X^{\\prime} 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:\n
X is a reflexive normed space (that is, is an isomorphism of normed spaces),
X is a reflexive locally convex space (that is, is an isomorphism of topological vector spaces ),
X is a semi-reflexive locally convex space (that is, is surjective).

\n\n

A (somewhat artificial) 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.\nThen the continuous dual of X and Y^{\\prime} are the same set of functionals, and bounded subsets of X (that is, weakly bounded subsets of Y) are norm-bounded, hence the Banach space Y^{\\prime} is the strong dual of X. Since Y is reflexive, the continuous dual of is equal to the image J(X) of X under the canonical embedding J, but the topology on X (the weak topology of Y) is not the strong topology that is equal to the norm topology of Y.

[Montel spaces](https://bliptext.com/articles/montel-space) are reflexive locally convex topological vector spaces.\nIn particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:\n

\n\n\n## Counter-examples\n\n# Other types of reflexivity\nA stereotype space, or polar reflexive space, is defined as a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on [totally bounded](https://bliptext.com/articles/totally-[bounded](https://bliptext.com/articles/bounded-set)-set) subsets (instead of [bounded](https://bliptext.com/articles/bounded-set) subsets) in the definition of dual space X^{\\prime}.\nMore precisely, a TVS X is called polar reflexive or stereotype if the evaluation map into the second dual space\nis an [isomorphism of topological vector spaces](https://bliptext.com/articles/tvs-isomorphism).\nHere the stereotype dual space X^\\star is defined as the space of continuous linear functionals X^{\\prime} endowed with the topology of uniform convergence on totally bounded sets in X (and the stereotype second dual space is the space dual to X^{\\star} in the same sense).\nIn contrast to the classical reflexive spaces the class ******Ste****** of stereotype spaces is very wide (it contains, in particular, all [Fréchet spaces](https://bliptext.com/articles/fr-chet-space) and thus, all [Banach spaces](https://bliptext.com/articles/banach-space)), it forms a [closed monoidal category](https://bliptext.com/articles/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.\nSimilarly, one can replace the class of bounded (and totally bounded) subsets in X in the definition of dual space X^{\\prime}, 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, 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****.\n\n## Citations\n\n## General references","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:41:54.940Z","updated_at":"2025-01-25T18:41:54.940Z","search_vector":"'reflex':1 'space':2","search_content":" \n \n\n# \nIn the area of mathematics known as [functional analysis](https://bliptext.com/articles/functional-analysis), a **reflexive space** is a [locally convex](https://bliptext.com/articles/locally-convex) [topological vector space](https://bliptext.com/articles/topological-vector-space) for which the canonical evaluation map from X into its [bidual](https://bliptext.com/articles/bidual) (which is the [strong dual](https://bliptext.com/articles/strong-dual) of the [strong dual](https://bliptext.com/articles/strong-dual) of X) is a [homeomorphism](https://bliptext.com/articles/homeomorphism) (or equivalently, a [TVS isomorphism](https://bliptext.com/articles/tvs-isomorphism)).\nA [normed space](https://bliptext.com/articles/normed-space) is reflexive if and only if this canonical evaluation map is [surjective](https://bliptext.com/articles/surjective), in which case this (always linear) evaluation map is an [isometric isomorphism](https://bliptext.com/articles/isometric-isomorphism) and the [normed space](https://bliptext.com/articles/normed-space) is a [Banach space](https://bliptext.com/articles/banach-space).\nThose spaces for which the canonical evaluation map is surjective are called [semi-reflexive](https://bliptext.com/articles/semi-reflexive) spaces.\nIn 1951, [R. C. James](https://bliptext.com/articles/robert-c-james) discovered a Banach space, now known as [James' space](https://bliptext.com/articles/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](https://bliptext.com/articles/isometric-isomorphism) is necessarily the canonical evaluation map).\nSo 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.\nReflexive spaces play an important role in the general theory of [locally convex](https://bliptext.com/articles/locally-convex) TVSs and in the theory of [Banach spaces](https://bliptext.com/articles/banach-space) in particular.\n[Hilbert spaces](https://bliptext.com/articles/hilbert-space) are prominent examples of reflexive Banach spaces.\nReflexive Banach spaces are often characterized by their geometric properties.\n\n# Definition\nSuppose that X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) over the field \\mathbb{F} (which is either the real or complex numbers) whose [continuous dual space](https://bliptext.com/articles/continuous-dual-space), X^{\\prime}, **separates points** on X (that is, for any there exists some such that ).\nLet (some texts write ) denote the [strong dual](https://bliptext.com/articles/strong-dual) of X, which is the vector space X^{\\prime} of continuous linear functionals on X endowed with the [topology of uniform convergence](https://bliptext.com/articles/topology-of-uniform-convergence) on [bounded subsets](https://bliptext.com/articles/topology-of-uniform-convergence) of X; this topology is also called the **[strong dual](https://bliptext.com/articles/strong-dual) topology** and it is the \"default\" topology placed on a continuous dual space (unless another topology is specified).\nIf X is a normed space, then the strong dual of X is the continuous dual space X^{\\prime} with its usual norm topology.\nThe **bidual** of X, denoted by is the strong dual of ; that is, it is the space\nIf X is a normed space, then is the continuous dual space of the Banach space with its usual norm topology.\nFor any x \\in X, let be defined by where J_x is a linear map called the **evaluation map at x**;\nsince is necessarily continuous, it follows that\nSince X^{\\prime} separates points on X, the linear map defined by J(x) := J_x is injective where this map is called the **evaluation map** or the **canonical map**.\nCall X **semi-**reflexive**** if is bijective (or equivalently, [surjective](https://bliptext.com/articles/surjective)) and we call X **reflexive** if in addition is an isomorphism of TVSs.\nA [normable](https://bliptext.com/articles/normable) space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.\n\n# Reflexive Banach spaces\nSuppose X is a [normed vector space](https://bliptext.com/articles/normed-vector-space) over the number field or (the [real numbers](https://bliptext.com/articles/real-number) or the [complex numbers](https://bliptext.com/articles/complex-number)), with a norm Consider its [[dual norm](https://bliptext.com/articles/dual-norm)ed space](https://bliptext.com/articles/dual-norm) X^{\\prime}, that consists of all [continuous](https://bliptext.com/articles/continuous-function) [linear functionals](https://bliptext.com/articles/linear-functional) and is equipped with the [dual norm](https://bliptext.com/articles/dual-norm) defined by\nThe dual X^{\\prime} is a normed space (a [Banach space](https://bliptext.com/articles/banach-space) to be precise), and its dual normed space is called **bidual space** for X. The bidual consists of all continuous linear functionals and is equipped with the norm dual to Each vector x \\in X generates a scalar function by the formula:\nand J(x) is a continuous linear functional on X^{\\prime}, that is, One obtains in this way a map\ncalled **evaluation map**, that is linear.\nIt follows from the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) that J is injective and preserves norms:\nthat is, J maps X isometrically onto its image J(X) in Furthermore, the image J(X) is closed in but it need not be equal to\nA normed space X is called **reflexive** if it satisfies the following equivalent conditions:\n the evaluation map is [surjective](https://bliptext.com/articles/bijection-injection-and-surjection), \n the evaluation map is an [isometric isomorphism](https://bliptext.com/articles/banach-space) of normed spaces, \n the evaluation map is an [isomorphism](https://bliptext.com/articles/banach-space) of normed spaces. \n \nA reflexive space X is a Banach space, since X is then isometric to the Banach space\n\n## Remark\nA Banach space X is reflexive if it is linearly isometric to its [bidual](https://bliptext.com/articles/dual-space) under this canonical embedding J. [James' space](https://bliptext.com/articles/james-space) is an example of a non-reflexive space which is linearly isometric to its [bidual](https://bliptext.com/articles/dual-space).\nFurthermore, the image of James' space under the canonical embedding J has [codimension](https://bliptext.com/articles/codimension) one in its bidual.\nA Banach space X is called **quasi-reflexive** (of order d) if the quotient has finite dimension d.\n\n## Examples\n\n## Properties\nSince every finite-dimensional normed space is a reflexive [Banach space](https://bliptext.com/articles/banach-space), only infinite-dimensional spaces can be non-reflexive.\nIf a Banach space Y is isomorphic to a reflexive Banach space X then Y is reflexive.\nEvery [closed](https://bliptext.com/articles/closed-set) [linear subspace](https://bliptext.com/articles/linear-subspace) of a reflexive space is reflexive.\nThe continuous dual of a reflexive space is reflexive.\nEvery [quotient](https://bliptext.com/articles/quotient-space-linear-algebra) of a reflexive space by a closed subspace is reflexive.\nLet X be a Banach space.\nThe following are equivalent.\n \n The space X is reflexive. \n The continuous dual of X is reflexive.\n \n The closed unit ball of X is [compact](https://bliptext.com/articles/compact-space) in the [weak topology](https://bliptext.com/articles/weak-topology).\n(This is known as Kakutani's Theorem.) \n Every bounded sequence in X has a weakly convergent subsequence.\n \n The statement of [Riesz's lemma](https://bliptext.com/articles/riesz-s-lemma) holds when the real number is exactly 1.\nExplicitly, for every closed proper vector subspace Y of X, there exists some vector u \\in X of unit norm \\|u\\| = 1 such that for all y \\in Y. \n Every continuous linear functional on X attains its supremum on the closed unit ball in X. ([James' theorem](https://bliptext.com/articles/james-theorem)) \n \nSince norm-closed [convex subsets](https://bliptext.com/articles/convex-set) in a Banach space are weakly closed,\nit follows from the third property that closed bounded convex subsets of a reflexive space X are weakly compact.\nThus, for every decreasing sequence of non-empty closed bounded convex subsets of X, the intersection is non-empty.\nAs a consequence, every continuous [convex function](https://bliptext.com/articles/convex-function) f on a closed convex subset C of X, such that the set\nis non-empty and bounded for some real number t, attains its minimum value on C.\nThe promised geometric property of reflexive Banach spaces is the following: if C is a closed non-empty [convex](https://bliptext.com/articles/convex-set) subset of the reflexive space X, then for every x \\in X there exists a c \\in C such that \\|x - c\\| minimizes the distance between x and points of C. This follows from the preceding result for [convex](https://bliptext.com/articles/convex-set) functions, applied to Note that while the minimal distance between x and C is uniquely defined by x, the point c is not.\nThe closest point c is unique when X is uniformly convex.\nA reflexive Banach space is [separable](https://bliptext.com/articles/separable-space) if and only if its continuous dual is [separable](https://bliptext.com/articles/separable-space).\nThis follows from the fact that for every normed space Y, separability of the continuous dual Y^{\\prime} implies separability of Y.\n\n## Super-reflexive space\nInformally, 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.\nBy this definition, the space X itself must be reflexive.\nAs an elementary example, every Banach space Y whose two dimensional subspaces are [isometric](https://bliptext.com/articles/isometry) to subspaces of X = \\ell^2 satisfies the [parallelogram law](https://bliptext.com/articles/parallelogram-law), hence\nY is a Hilbert space, therefore Y is reflexive.\nSo \\ell^2 is super-reflexive.\nThe formal definition does not use isometries, but almost isometries.\nA Banach space Y is **finitely representable**\nin a Banach space X if for every finite-dimensional subspace Y_0 of Y and every there is a subspace X_0 of X such that the multiplicative [Banach–Mazur distance](https://bliptext.com/articles/banach-mazur-compactum) between X_0 and Y_0 satisfies\nA Banach space finitely representable in \\ell^2 is a Hilbert space.\nEvery Banach space is finitely representable in c_0.\nThe [Lp space](https://bliptext.com/articles/lp-space) L^p([0, 1]) is finitely representable in \\ell^p.\nA 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](https://bliptext.com/articles/ultraproduct) of a family of Banach spaces\nallows for a concise definition: the Banach space X is super-reflexive when its ultrapowers are reflexive.\nJames proved that a space is super-reflexive if and only if its dual is super-reflexive.\n\n## Finite trees in Banach spaces\nOne of James' characterizations of super-reflexivity uses the growth of separated trees.\nThe description of a vectorial binary tree begins with a [[root](https://bliptext.com/articles/tree-graph-theory)ed binary tree](https://bliptext.com/articles/[root](https://bliptext.com/articles/tree-graph-theory)ed-binary-tree) labeled by vectors: a tree of [height](https://bliptext.com/articles/tree-graph-theory) n in a Banach space X is a family of 2^{n+1} - 1 vectors of X, that can be organized in successive levels, starting with level 0 that consists of a single vector the [root](https://bliptext.com/articles/tree-graph-theory) of the tree, followed, for by a family of s^k2 vectors forming level k:\nthat are the [children](https://bliptext.com/articles/tree-graph-theory) of vertices of level k - 1.\nIn addition to the [tree structure](https://bliptext.com/articles/tree-graph-theory), it is required here that each vector that is an [internal vertex](https://bliptext.com/articles/tree-graph-theory) of the tree be the midpoint between its two children:\nGiven 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:\n**Theorem.**\nThe Banach space X is super-reflexive if and only if for every there is a number n(t) such that every t-separated tree contained in the unit ball of X has height less than n(t).\n[Uniformly convex spaces](https://bliptext.com/articles/uniformly-convex-space) are super-reflexive.\nLet X be uniformly convex, with [modulus of convexity](https://bliptext.com/articles/modulus-and-characteristic-of-convexity) \\delta_X and let t be a real number in (0, 2]. By the [properties](https://bliptext.com/articles/modulus-and-characteristic-of-convexity) of the [modulus of convexity](https://bliptext.com/articles/modulus-and-characteristic-of-convexity), a t-separated tree of height n, contained in the unit ball, must have all points of level n - 1 contained in the ball of radius By induction, it follows that all points of level n - k are contained in the ball of radius\nIf the height n was so large that\nthen the two points x_1, x_{-1} of the first level could not be t-separated, contrary to the assumption.\nThis gives the required bound n(t), function of \\delta_X(t) only.\nUsing the tree-characterization, [Enflo](https://bliptext.com/articles/per-enflo) proved\nthat super-reflexive Banach spaces admit an equivalent uniformly convex norm.\nTrees in a Banach space are a special instance of vector-valued [martingales](https://bliptext.com/articles/martingale-probability-theory).\nAdding techniques from scalar martingale theory, [Pisier](https://bliptext.com/articles/gilles-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 c > 0 and some real number q \\geq 2,\n\n# Reflexive locally convex spaces\nThe notion of reflexive Banach space can be generalized to [topological vector spaces](https://bliptext.com/articles/topological-vector-space) in the following way.\nLet X be a topological vector space over a number field \\mathbb F (of [real numbers](https://bliptext.com/articles/real-number) \\mathbb R or [complex numbers](https://bliptext.com/articles/complex-number) \\Complex).\nConsider its [strong dual space](https://bliptext.com/articles/strong-topology-polar-topology) which consists of all [continuous](https://bliptext.com/articles/continuous-function) [linear functionals](https://bliptext.com/articles/linear-functional) and is equipped with the [strong topology](https://bliptext.com/articles/strong-topology-polar-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](https://bliptext.com/articles/strong-topology-polar-topology) which is called the **strong bidual space** for X. It consists of all [continuous](https://bliptext.com/articles/continuous-function) [linear functionals](https://bliptext.com/articles/linear-functional) and is equipped with the [strong topology](https://bliptext.com/articles/strong-topology-polar-topology) Each vector x \\in X generates a map by the following formula:\nThis is a continuous linear functional on that is,, This induces a map called the **evaluation map**:\nThis map is linear.\nIf X is locally convex, from the [Hahn–Banach theorem](https://bliptext.com/articles/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 ).\nBut it can be non-surjective and/or discontinuous.\nA locally convex space X is called\n\n## Semireflexive spaces\n\n### Characterizations\nIf X is a Hausdorff locally convex space then the following are equivalent:\n\n## Characterizations of reflexive spaces\nIf X is a Hausdorff locally convex space then the following are equivalent:\nIf X is a normed space then the following are equivalent:\n\n## Sufficient conditions\nA normed space that is semireflexive is a reflexive Banach space.\nA closed vector subspace of a reflexive Banach space is reflexive.\nLet X be a Banach space and M a closed vector subspace of X. If two of X, M, and X / M are reflexive then they all are.\nThis is why reflexivity is referred to as a.\nIf a [barreled](https://bliptext.com/articles/barreled-space) locally convex Hausdorff space is semireflexive then it is reflexive.\nThe strong dual of a reflexive space is reflexive.Every [Montel space](https://bliptext.com/articles/montel-space) is reflexive.\nAnd the strong dual of a [Montel space](https://bliptext.com/articles/montel-space) is a [Montel space](https://bliptext.com/articles/montel-space) (and thus is reflexive).\n\n## Properties\nA locally convex Hausdorff reflexive space is [barrelled](https://bliptext.com/articles/barrelled-space).\nIf X is a normed space then is an isometry onto a closed subspace of This isometry can be expressed by:\nSuppose that X is a normed space and is its bidual equipped with the bidual norm.\nThen the unit ball of X, is dense in the unit ball\nof for the weak topology\n\n## Examples\n \n Every finite-dimensional Hausdorff [topological vector space](https://bliptext.com/articles/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. \n A normed space X is reflexive as a normed space if and only if it is reflexive as a locally convex space.\nThis follows from the fact that for a normed space X its dual normed space X^{\\prime} 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:\n X is a reflexive normed space (that is, is an isomorphism of normed spaces), \n X is a reflexive locally convex space (that is, is an isomorphism of topological vector spaces ), \n X is a semi-reflexive locally convex space (that is, is surjective). \n \n \n A (somewhat artificial) 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.\nThen the continuous dual of X and Y^{\\prime} are the same set of functionals, and bounded subsets of X (that is, weakly bounded subsets of Y) are norm-bounded, hence the Banach space Y^{\\prime} is the strong dual of X. Since Y is reflexive, the continuous dual of is equal to the image J(X) of X under the canonical embedding J, but the topology on X (the weak topology of Y) is not the strong topology that is equal to the norm topology of Y. \n [Montel spaces](https://bliptext.com/articles/montel-space) are reflexive locally convex topological vector spaces.\nIn particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:\n \n \n\n## Counter-examples\n\n# Other types of reflexivity\nA stereotype space, or polar reflexive space, is defined as a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on [totally [bounded](https://bliptext.com/articles/bounded-set)](https://bliptext.com/articles/totally-[bounded](https://bliptext.com/articles/bounded-set)-set) subsets (instead of [bounded](https://bliptext.com/articles/bounded-set) subsets) in the definition of dual space X^{\\prime}.\nMore precisely, a TVS X is called polar reflexive or stereotype if the evaluation map into the second dual space\nis an [isomorphism of topological vector spaces](https://bliptext.com/articles/tvs-isomorphism).\nHere the stereotype dual space X^\\star is defined as the space of continuous linear functionals X^{\\prime} endowed with the topology of uniform convergence on totally bounded sets in X (and the stereotype second dual space is the space dual to X^{\\star} in the same sense).\nIn contrast to the classical reflexive spaces the class ******Ste****** of stereotype spaces is very wide (it contains, in particular, all [Fréchet spaces](https://bliptext.com/articles/fr-chet-space) and thus, all [Banach spaces](https://bliptext.com/articles/banach-space)), it forms a [closed monoidal category](https://bliptext.com/articles/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.\nSimilarly, one can replace the class of bounded (and totally bounded) subsets in X in the definition of dual space X^{\\prime}, 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, 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****.\n\n## Citations\n\n## General references","revisionCount":0},"hasMore":false,"nextCursor":null}

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