Reflexive space

If X is a Hausdorff locally convex space then the following are equivalent:

1. X is semireflexive;
2. the weak topology on X had the Heine-Borel property (i.e. for the weak topology ${\displaystyle \sigma \left(X,X^{\prime }\right)}$, every closed and bounded subset of ${\displaystyle X_{\sigma }}$ is weakly compact).[1]
3. If linear form on ${\displaystyle X^{\prime }}$ that continuous when ${\displaystyle X^{\prime }}$ has the strong dual topology, then it is continuous when ${\displaystyle X^{\prime }}$ has the weak topology;[2]
4. ${\displaystyle X_{\tau }^{\prime }}$ is barreled;[2]
5. X weak the weak topology ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ is quasi-complete.[2]

If X is a Hausdorff locally convex space then the following are equivalent:

1. X is reflexive;
2. X is semireflexive and infrabarreled;[3]
3. X is semireflexive and barreled;
4. X is barreled and the weak topology on X had the Heine-Borel property (i.e. for the weak topology ${\displaystyle \sigma \left(X,X^{\prime }\right)}$, every closed and bounded subset of ${\displaystyle X_{\sigma }}$ is weakly compact).[1]
5. X is semireflexive and quasibarrelled.[4]

If X is a normed space then the following are equivalent:

1. X is reflexive;
2. the closed unit ball is compact when X has the weak topology ${\displaystyle \sigma \left(X,X^{\prime }\right)}$.[5]
3. X is a Banach space and ${\displaystyle X_{b}^{\prime }}$ is reflexive.[6]
4. Every sequence ${\displaystyle \{C_{n}\}_{n=1}^{\infty }}$, with ${\displaystyle C_{n+1}\subset C_{n}\forall n}$, of nonempty closed bounded convex subsets of X has nonempty intersection.[7]

Theorem:[8] 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.

James' theorem: A Banach space B is reflexive if and only if every continuous linear functional on B attains its supremum on the closed unit ball in B.

• A closed vector subspace of a reflexive Banach space is reflexive.[3]
• Let 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.[3]
  • This is why reflexivity is referred to as a three-space property.[3]
• The strong dual of a reflexive space is reflexive.[9]
• If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.[1]
• A normed space that is semireflexive is a reflexive Banach space.[10]
• Every Montel space is reflexive.[5]
• The strong dual of a Montel space is a Montel space (and thus is reflexive).[5]

• There exists a non-reflexive locally convex TVS whose strong dual is reflexive.[11]

• A locally convex Hausdorff reflexive space is barrelled.
• If X is a normed space then ${\displaystyle I:X\to X^{\prime \prime }}$ is an isometry onto a closed subspace of ${\displaystyle X^{\prime \prime }}$.[10] This isometry can be expressed by:

  ${\displaystyle \|x\|=\sup _{x^{\prime }\in X^{\prime },\|x^{\prime }\|\leq 1}|\langle x^{\prime },x\rangle |}$.

• Suppose that X is a normed space and ${\displaystyle X^{\prime \prime }}$ is its bidual equipped with the bidual norm. Then the unit ball of X, ${\displaystyle I\left(\{x\in X:\|x\|\leq 1\}\right)}$ is dense in the unit ball ${\displaystyle \left\{x^{\prime \prime }\in X^{\prime \prime }:\|x^{\prime \prime }\|\leq 1\right\}}$ of ${\displaystyle X^{\prime \prime }}$ for the weak topology ${\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)}$.[10]

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. [12] A Banach space X is called quasi-reflexive (of order d) if the quotient X ′′ / J(X) has finite dimension d.

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.

If a Banach space Y is isomorphic to a reflexive Banach space X, then Y is reflexive.[13]

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.[14]

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

1. The space X is reflexive.
2. The continuous dual of X is reflexive.[15]
3. The closed unit ball of X is compact in the weak topology. (This is known as Kakutani's Theorem.)[16]
4. Every bounded sequence in X has a weakly convergent subsequence.[17]
5. Every continuous linear functional on X attains its maximum on the closed unit ball in X.[18] (James' theorem)

Since norm-closed convex subsets in a Banach space are weakly closed,[19] 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

${\displaystyle C_{t}=\{x\in C\,:\,f(x)\leq t\}}$

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.[20]

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[21] 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[22] 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

${\displaystyle d(X_{0},Y_{0})<1+\varepsilon .}$

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[23] 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.[22]

One of James' characterizations of super-reflexivity uses the growth of separated trees.[24] 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:

${\displaystyle \{x_{\varepsilon _{1},\ldots ,\varepsilon _{k}}\},\quad \varepsilon _{j}=\pm 1,\quad j=1,\ldots ,k,}$

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:

${\displaystyle x_{\emptyset }={\frac {x_{1}+x_{-1}}{2}},\quad x_{\varepsilon _{1},\ldots ,\varepsilon _{k}}={\frac {x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}+x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}}{2}},\quad 1\leq k<n.}$

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:

${\displaystyle \|x_{1}-x_{-1}\|\geq t,\quad \|x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}-x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}\|\geq t,\quad 1\leq k<n.}$

Theorem.[24] 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.[24] 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

${\displaystyle (1-\delta _{X}(t))^{j},\ j=1,\ldots ,n.}$

If the height n was so large that

${\displaystyle (1-\delta _{X}(t))^{n-1}<t/2,}$

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[25] 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[26] 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,

${\displaystyle \delta _{X}(t)\geq c\,t^{q},\quad t\in [0,2].}$

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 ${\displaystyle X}$ 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 ${\displaystyle X}$ its dual normed space ${\displaystyle X'}$ coincides as a topological vector space with the strong dual space ${\displaystyle X'_{\beta }}$. As a corollary, the evaluation map ${\displaystyle J:X\to X''}$ coincides with the evaluation map ${\displaystyle J:X\to (X'_{\beta })'_{\beta }}$, and the following conditions become equivalent:

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

(ii) ${\displaystyle X}$ is a reflexive locally convex space (i.e. ${\displaystyle J:X\to (X'_{\beta })'_{\beta }}$ is an isomorphism of topological vector spaces[27]),

(iii) ${\displaystyle X}$ is a semi-reflexive locally convex space (i.e. ${\displaystyle J:X\to (X'_{\beta })'_{\beta }}$ 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:[32]

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