Reflexive space

In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space. Reflexive Banach spaces are often characterized by their geometric properties.

Reflexive Banach spaces

Suppose $X$ is a normed vector space over the number field $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb {C}$ (the real or complex numbers), with a norm $\|\cdot\|$. Consider its dual normed space $X'$, that consists of all continuous linear functionals $f:X\to {\mathbb F}$ and is equipped with the dual norm $\|\cdot\|'$ defined by

$\|f\|' = \sup \{ |f(x)| \,:\, x \in X, \ \|x\| \le 1 \}.$

The dual $X'$ is a normed space (a Banach space to be precise), and its dual normed space $X''=(X')'$ is called bidual space for $X$. The bidual consists of all continuous linear functionals $h:X'\to {\mathbb F}$ and is equipped with the norm $\|\cdot\|''$ dual to $\|\cdot\|'$. Each vector $x\in X$ generates a scalar function $J(x):X'\to{\mathbb F}$ by the formula:

$J(x)(f)=f(x),\qquad f\in X',$

and $J(x)$ is a continuous linear functional on $X'$, i.e., $J(x)\in X''$. One obtains in this way a map

$J: X \to X''$

called evaluation map, that is linear. It follows from the Hahn–Banach theorem that $J$ is injective and preserves norms:

$\forall x\in X\qquad \|J(x)\|''=\|x\|,$

i.e., $J$ maps $X$ isometrically onto its image $J(X)$ in $X''$. Furthermore, the image $J(X)$ is closed in $X''$, but it need not be equal to $X''$.

A normed space $X$ is called reflexive if it satisfies the following equivalent conditions:

(i) the evaluation map $J:X\to X''$ is surjective,

(ii) the evaluation map $J:X\to X''$ is an isometric isomorphism of normed spaces,

(iii) the evaluation map $J:X\to X''$ is an isomorphism of normed spaces.

A reflexive space $X$ is a Banach space, since $X$ is then isometric to the Banach space $X''$.

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. ^[1] A Banach space X is called quasi-reflexive (of order d) if the quotient X ′′ / J(X) 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 L^p spaces for 1 < p < ∞. More generally: all uniformly convex Banach spaces are reflexive according to the Milman–Pettis theorem. The L¹(μ) and L^∞(μ) spaces are not reflexive (unless they are finite dimensional, which happens for example when μ is a measure on a finite set). Likewise, the Banach space C([0, 1]) of continuous functions on [0, 1] is not reflexive.

4) The spaces S_p(H) of operators in the Schatten class on a Hilbert space H are uniformly convex, hence reflexive, when 1 < p < ∞. When the dimension of H is infinite, then S₁(H) (the trace class) is not reflexive, because it contains a subspace isomorphic to ℓ¹, and S_∞(H) = L(H) (the bounded linear operators on H) 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.^[2]

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.^[3]

Let X be a Banach space. The following are equivalent.

1. The space X is reflexive.
2. The continuous dual of X is reflexive.^[4]
3. The closed unit ball of X is compact in the weak topology. (This is known as Kakutani's Theorem.)^[5]
4. Every bounded sequence in X has a weakly convergent subsequence.^[6]
5. Every continuous linear functional on X attains its maximum on the closed unit ball in X.^[7] (James' theorem)

Since norm-closed convex subsets in a Banach space are weakly closed,^[8] 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

$C_t = \{ x \in C \,:\, f(x) \le t\}$

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 ǁx − cǁ minimizes the distance between x and points of C. This follows from the preceding result for convex functions, applied to f(y) = ǁy − xǁ. 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 Y ′ implies separability of Y.^[9]

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 X = ℓ² satisfies the parallelogram law, hence^[10] 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^[11] in a Banach space X if for every finite-dimensional subspace Y₀ of Y and every ε > 0, there is a subspace X₀ of X such that the multiplicative Banach–Mazur distance between X₀ and Y₀ satisfies

$d(X_0, Y_0) < 1 + \varepsilon.$

A Banach space finitely representable in ℓ² is a Hilbert space. Every Banach space is finitely representable in c₀. The space L^p([0, 1]) is finitely representable in ℓ^p.

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^[12] 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.^[11]

Finite trees in Banach spaces

One of James' characterizations of super-reflexivity uses the growth of separated trees.^[13] 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 2^n + 1 − 1 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 k = 1, …, n, by a family of 2^k vectors forming level k:

$\{x_{\varepsilon_1, \ldots, \varepsilon_k}\}, \quad \varepsilon_j = \pm 1, \quad j = 1, \ldots, k,$

that are the children of vertices of level k − 1. 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:

$x_\emptyset = \frac{x_1 + x_{-1}}{2}, \quad x_{\varepsilon_1, \ldots, \varepsilon_k} = \frac{x_{\varepsilon_1, \ldots, \varepsilon_k, 1} + x_{\varepsilon_1, \ldots, \varepsilon_k, -1}} {2}, \quad 1 \le k < n.$

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:

$\|x_1 - x_{-1}\| \ge t, \quad \|x_{\varepsilon_1, \ldots, \varepsilon_k, 1} - x_{\varepsilon_1, \ldots, \varepsilon_k, -1}\| \ge t, \quad 1 \le k < n.$

Theorem.^[13] The Banach space X is super-reflexive if and only if for every t ∈ (0, 2], 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).

Uniformly convex spaces are super-reflexive.^[13] Let X be uniformly convex, with modulus of convexity δ_X and let t be a real number in (0, 2]. 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 n − 1 contained in the ball of radius 1 − δ_X(t) < 1. By induction, it follows that all points of level n − j are contained in the ball of radius

$(1 - \delta_X(t))^{j}, \ j = 1, \ldots, n.$

If the height n was so large that

$(1 - \delta_X(t))^{n-1} < t / 2,$

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(t), function of δ_X(t) only.

Using the tree-characterization, Enflo proved^[14] 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^[15] 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 ≥ 2,

$\delta_X(t) \ge c \, t^q, \quad t \in [0, 2].$

Reflexive locally convex spaces

The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.

Let $X$ be a topological vector space over a number field $\mathbb F$ (of real numbers $\mathbb R$ or complex numbers $\mathbb C$). Consider its strong dual space $X'_\beta$, which consists of all continuous linear functionals $f:X\to {\mathbb F}$ and is equipped with the strong topology $\beta(X',X)$, i.e., the topology of uniform convergence on bounded subsets in $X$. The space $X'_\beta$ is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space $(X'_\beta)'_\beta$, which is called the strong bidual space for $X$. It consists of all continuous linear functionals $h:X'_\beta\to {\mathbb F}$ and is equipped with the strong topology $\beta((X'_\beta)',X'_\beta)$. Each vector $x\in X$ generates a map $J(x):X'_\beta\to{\mathbb F}$ by the following formula:

$J(x)(f)=f(x),\qquad f\in X'.$

This is a continuous linear functional on $X'_\beta$, i.e., $J(x)\in (X'_\beta)'_\beta$. One obtains a map called evaluation map:

$J:X\to (X'_\beta)'_\beta.$

This map is linear. If $X$ is locally convex, from the Hahn-Banach theorem it follows that $J$ is injective and open (i.e., for each neighbourhood of zero $U$ in $X$ there is a neighbourhood of zero $V$ in $(X'_\beta)'_\beta$ such that $J(U)\supseteq V\cap J(X)$). But it can be non-surjective and/or discontinuous.

A locally convex space $X$ is called

- semi-reflexive if the evaluation map $J:X\to (X'_\beta)'_\beta$ is surjective,

- reflexive if the evaluation map $J:X\to (X'_\beta)'_\beta$ is surjective and continuous (in this case $J$ is an isomorphism of topological vector spaces).

Theorem. A locally convex Hausdorff space $X$ is semi-reflexive if and only if $X$ with the $\sigma(X, X^*)$-topology has the Heine-Borel property (i.e. weakly closed and bounded subsets of $X$ are weakly compact).

Theorem.^[16] A locally convex space $X$ 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 $X$ 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 $X$ its dual normed space $X'$ coincides as a topological vector space with the strong dual space $X'_\beta$. As a corollary, the evaluation map $J:X\to X''$ coincides with the evaluation map $J:X\to (X'_\beta)'_\beta$, and the following conditions become equivalent:

(i) $X$ is a reflexive normed space (i.e. $J:X\to X''$ is an isomorphism of normed spaces),

(ii) $X$ is a reflexive locally convex space (i.e. $J:X\to (X'_\beta)'_\beta$ is an isomorphism of topological vector spaces),

(iii) $X$ is a semi-reflexive locally convex space (i.e. $J:X\to (X'_\beta)'_\beta$ is surjective).

3) 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 (Y, σ(Y, Y ′)), that is, the vector space Y equipped with the weak topology. Then the continuous dual of X and Y ′ are the same set of functionals, and bounded subsets of X (i.e., weakly bounded subsets of Y) are norm-bounded, hence the Banach space Y ′ is the strong dual of X. Since Y is reflexive, the continuous dual of X ′ = Y ′ 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 β(X, X ′), 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:^[17]

• the space $C^\infty(M)$ of smooth functions on arbitrary (real) smooth manifold $M$, and its strong dual space $(C^\infty)'(M)$ of distributions with compact support on $M$,
• the space ${\mathcal D}(M)$ of smooth functions with compact support on arbitrary (real) smooth manifold $M$, and its strong dual space ${\mathcal D}'(M)$ of distributions on $M$,
• the space ${\mathcal O}(M)$ of holomorphic functions on arbitrary complex manifold $M$, and its strong dual space ${\mathcal O}'(M)$ of analytic functionals on $M$,
• the Schwartz space ${\mathcal S}({\mathbb R}^n)$ on ${\mathbb R}^n$, and its strong dual space ${\mathcal S}'({\mathbb R}^n)$ of tempered distributions on ${\mathbb R}^n$.

Stereotype spaces and other versions of reflexivity

Among all locally convex spaces (even among all Banach spaces) used in functional analysis the class of reflexive spaces is too narrow to represent a self-sufficient category in any sense. On the other hand, the idea of duality reflected in this notion is so natural that it gives rise to intuitive expectations that appropriate changes in the definition of reflexivity can lead to another notion, more convenient for some goals of mathematics. One of such goals is the idea of approaching analysis to the other parts of mathematics, like algebra and geometry, by reformulating its results in the purely algebraic language of category theory.

This program is being worked out in the theory of stereotype spaces, which are defined as topological vector spaces 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 X’. More precisely, a topological vector space $X$ is called stereotype if the evaluation map into the stereotype second dual space

$J:X\to X^{\star\star},\quad J(x)(f)=f(x),\quad x\in X,\quad f\in X^\star$

is an isomorphism of topological vector spaces. Here the stereotype dual space $X^\star$ is defined as the space of continuous linear functionls $X'$ endowed with the topology of uniform convergence totally bounded sets in $X$ (and the stereotype second dual space $X^{\star\star}$ is the space dual to $X^{\star}$ 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.

Similarly, one can replace the class of bounded (and totally 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,^[18][19] 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.
• Reflexive operator algebra

Notes

1. R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37: 174–177. doi:10.1073/pnas.37.3.174.
2. Proposition 1.11.8, p. 99 in Megginson (1998).
3. pp. 104–105 in Megginson (1998).
4. Corollary 1.11.17, p. 104 in Megginson (1998).
5. Conway, Theorem V.4.2, p. 135.
6. Since weak compactness and weak sequential compactness coincide by the Eberlein–Šmulian theorem.
7. Theorem 1.13.11, p. 125 in Megginson (1998).
8. Theorem 2.5.16, p. 216 in Megginson (1998).
9. Theorem 1.12.11, p. 112 and Corollary 1.12.12, p. 113 in Megginson (1998).
10. see this characterization of Hilbert space among Banach spaces
11. James, Robert C. (1972), "Super-reflexive Banach spaces", Canad. J. Math. 24:896–904.
12. Dacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math. 41:315–334.
13. see James (1972).
14. Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel J. Math. 13:281–288.
15. Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel J. Math. 20:326–350.
16. Schaefer (1966, 5.6, 5.5)
17. Edwards (1965, 8.4.7).
18. Garibay Bonales, F.; Trigos-Arrieta, F.J.; Vera Mendoza, R. (2002). "A characterization of Pontryagin-van Kampen duality for locally convex spaces". Topology and its Applications 121: 75–89. doi:10.1016/s0166-8641(01)00111-0.
19. Akbarov, S. S.; Shavgulidze, E. T. (2003). "On two classes of spaces reflexive in the sense of Pontryagin". Mat. Sbornik 194 (10): 3–26.

References

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