Locally convex topological vector space
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 formula_1 into its bidual (which is the strong dual of the strong dual of formula_1) 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 not 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 not 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.
Suppose that formula_1 is a topological vector space (TVS) over the field formula_4 (which is either the real or complex numbers) whose continuous dual space, formula_5 separates points on formula_1 (that is, for any formula_7 there exists some formula_8 such that formula_9). 
Let formula_10 (some texts write formula_11) denote the strong dual of formula_12 which is the vector space formula_13 of continuous linear functionals on formula_1 endowed with the topology of uniform convergence on bounded subsets of formula_1; 
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 formula_1 is a normed space, then the strong dual of formula_1 is the continuous dual space formula_13 with its usual norm topology. 
The bidual of formula_12 denoted by formula_20 is the strong dual of formula_10; that is, it is the space formula_22 
If formula_1 is a normed space, then formula_24 is the continuous dual space of the Banach space formula_10 with its usual norm topology. 
For any formula_26 let formula_27 be defined by formula_28 where formula_29 is a linear map called the evaluation map at formula_30; 
since formula_31 is necessarily continuous, it follows that formula_32 
Since formula_13 separates points on formula_12 the linear map formula_35 defined by formula_36 is injective where this map is called the evaluation map or the canonical map. 
Call formula_1 semi-reflexive if formula_35 is bijective (or equivalently, surjective) and we call formula_1 reflexive if in addition formula_40 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 formula_1 is a normed vector space over the number field formula_42 or formula_43 (the real numbers or the complex numbers), with a norm formula_44 Consider its dual normed space formula_5 that consists of all continuous linear functionals formula_46 and is equipped with the dual norm formula_47 defined by
formula_48
The dual formula_13 is a normed space (a Banach space to be precise), and its dual normed space formula_50 is called bidual space for formula_51 The bidual consists of all continuous linear functionals formula_52 and is equipped with the norm formula_53 dual to formula_54 Each vector formula_55 generates a scalar function formula_56 by the formula:
formula_57
and formula_58 is a continuous linear functional on formula_5 that is, formula_60 One obtains in this way a map
formula_61
called evaluation map, that is linear. It follows from the Hahn–Banach theorem that formula_62 is injective and preserves norms:
formula_63
that is, formula_62 maps formula_1 isometrically onto its image formula_66 in formula_67 Furthermore, the image formula_66 is closed in formula_20 but it need not be equal to formula_67
A normed space formula_1 is called reflexive if it satisfies the following equivalent conditions:
A reflexive space formula_1 is a Banach space, since formula_1 is then isometric to the Banach space formula_67
Remark.
A Banach space formula_1 is reflexive if it is linearly isometric to its bidual under this canonical embedding formula_76 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 formula_62 has codimension one in its bidual.
A Banach space formula_1 is called quasi-reflexive (of order formula_79) if the quotient formula_80 has finite dimension formula_81
Properties.
Since every finite-dimensional normed space is a reflexive Banach space, only infinite-dimensional spaces can be non-reflexive.
If a Banach space formula_107 is isomorphic to a reflexive Banach space formula_1 then formula_107 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 formula_1 be a Banach space. The following are equivalent.
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 formula_1 are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of formula_12 the intersection is non-empty. As a consequence, every continuous convex function formula_113 on a closed convex subset formula_114 of formula_12 such that the set
formula_116
is non-empty and bounded for some real number formula_117 attains its minimum value on formula_118
The promised geometric property of reflexive Banach spaces is the following: if formula_114 is a closed non-empty convex subset of the reflexive space formula_12 then for every formula_55 there exists a formula_122 such that formula_123 minimizes the distance between formula_30 and points of formula_118 This follows from the preceding result for convex functions, applied toformula_126 Note that while the minimal distance between formula_30 and formula_114 is uniquely defined by formula_129 the point formula_130 is not. The closest point formula_130 is unique when formula_1 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 formula_133 separability of the continuous dual formula_134 implies separability of formula_135
Super-reflexive space.
Informally, a super-reflexive Banach space formula_1 has the following property: given an arbitrary Banach space formula_133 if all finite-dimensional subspaces of formula_107 have a very similar copy sitting somewhere in formula_12 then formula_107 must be reflexive. By this definition, the space formula_1 itself must be reflexive. As an elementary example, every Banach space formula_107 whose two dimensional subspaces are isometric to subspaces of formula_143 satisfies the parallelogram law, hence 
formula_107 is a Hilbert space, therefore formula_107 is reflexive. So formula_146 is super-reflexive.
The formal definition does not use isometries, but almost isometries. A Banach space formula_107 is finitely representable 
in a Banach space formula_1 if for every finite-dimensional subspace formula_149 of formula_107 and every formula_151 there is a subspace formula_152 of formula_1 such that the multiplicative Banach–Mazur distance between formula_152 and formula_149 satisfies
formula_156
A Banach space finitely representable in formula_146 is a Hilbert space. Every Banach space is finitely representable in formula_158 The Lp space formula_159 is finitely representable in formula_160
A Banach space formula_1 is super-reflexive if all Banach spaces formula_107 finitely representable in formula_1 are reflexive, or, in other words, if no non-reflexive space formula_107 is finitely representable in formula_51 The notion of ultraproduct of a family of Banach spaces 
allows for a concise definition: the Banach space formula_1 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 formula_167 in a Banach space formula_1 is a family of formula_169 vectors of formula_12 that can be organized in successive levels, starting with level 0 that consists of a single vector formula_171 the root of the tree, followed, for formula_172 by a family of formula_1732 vectors forming level formula_174
formula_175
that are the children of vertices of level formula_176 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:
formula_177
Given a positive real number formula_117 the tree is said to be formula_179-separated if for every internal vertex, the two children are formula_179-separated in the given space norm:
formula_181
Theorem.
The Banach space formula_1 is super-reflexive if and only if for every formula_183 there is a number formula_184 such that every formula_179-separated tree contained in the unit ball of formula_1 has height less than formula_187
Uniformly convex spaces are super-reflexive.
Let formula_1 be uniformly convex, with modulus of convexity formula_189 and let formula_179 be a real number in formula_191 By the properties of the modulus of convexity, a formula_179-separated tree of height formula_193 contained in the unit ball, must have all points of level formula_194 contained in the ball of radius formula_195 By induction, it follows that all points of level formula_196 are contained in the ball of radius
formula_197
If the height formula_167 was so large that 
formula_199
then the two points formula_200 of the first level could not be formula_179-separated, contrary to the assumption. This gives the required bound formula_202 function of formula_203 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 formula_1 admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant formula_205 and some real number formula_206
formula_207
Reflexive locally convex spaces.
The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.
Let formula_1 be a topological vector space over a number field formula_209 (of real numbers formula_210 or complex numbers formula_211). Consider its strong dual space formula_212 which consists of all continuous linear functionals formula_46 and is equipped with the strong topology formula_214 that is, the topology of uniform convergence on bounded subsets in formula_51 The space formula_10 is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space formula_217 which is called the strong bidual space for formula_51 It consists of all continuous linear functionals formula_219 and is equipped with the strong topology formula_220 Each vector formula_55 generates a map formula_222 by the following formula:
formula_223
This is a continuous linear functional on formula_212 that is, formula_225 This induces a map called the evaluation map:
formula_226
This map is linear. If formula_1 is locally convex, from the Hahn–Banach theorem it follows that formula_62 is injective and open (that is, for each neighbourhood of zero formula_229 in formula_1 there is a neighbourhood of zero formula_231 in formula_232 such that formula_233). But it can be non-surjective and/or discontinuous.
A locally convex space formula_1 is called 
 ;templatestyles src="Math_theorem/styles.css" / ;
Theorem — A locally convex Hausdorff space formula_1 is semi-reflexive if and only if formula_1 with the formula_240-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of formula_1 are weakly compact).
 ;templatestyles src="Math_theorem/styles.css" / ;
Theorem — A locally convex space formula_1 is reflexive if and only if it is semi-reflexive and barreled.
 ;templatestyles src="Math_theorem/styles.css" / ;
Theorem — The strong dual of a semireflexive space is barrelled.
 ;templatestyles src="Math_theorem/styles.css" / ;
Theorem — 
If formula_1 is a Hausdorff locally convex space then the canonical injection from formula_1 into its bidual is a topological embedding if and only if formula_1 is infrabarreled.
Semireflexive spaces.
Characterizations.
If formula_1 is a Hausdorff locally convex space then the following are equivalent:
Characterizations of reflexive spaces.
If formula_1 is a Hausdorff locally convex space then the following are equivalent:
If formula_1 is a normed space then the following are equivalent:
 ;templatestyles src="Math_theorem/styles.css" / ;
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.
 ;templatestyles src="Math_theorem/styles.css" / ;
James' theorem — 
A Banach space formula_276 is reflexive if and only if every continuous linear functional on formula_276 attains its supremum on the closed unit ball in formula_278
Sufficient conditions.
A normed space that is semireflexive is a reflexive Banach space.
A closed vector subspace of a reflexive Banach space is reflexive.
Let formula_1 be a Banach space and formula_280 a closed vector subspace