Reflexive space

Summary

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 into its bidual (which is the strong dual of the strong dual of ) 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 edit

Definition of the bidual

Suppose that   is a topological vector space (TVS) over the field   (which is either the real or complex numbers) whose continuous dual space,   separates points on   (that is, for any   there exists some   such that  ). Let   (some texts write  ) denote the strong dual of   which is the vector space   of continuous linear functionals on   endowed with the topology of uniform convergence on bounded subsets of  ; 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   is a normed space, then the strong dual of   is the continuous dual space   with its usual norm topology. The bidual of   denoted by   is the strong dual of  ; that is, it is the space  [1] If   is a normed space, then   is the continuous dual space of the Banach space   with its usual norm topology.

Definitions of the evaluation map and reflexive spaces

For any   let   be defined by   where   is a linear map called the evaluation map at  ; since   is necessarily continuous, it follows that   Since   separates points on   the linear map   defined by   is injective where this map is called the evaluation map or the canonical map. Call   semi-reflexive if   is bijective (or equivalently, surjective) and we call   reflexive if in addition   is an isomorphism of TVSs.[1] 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 edit

Suppose   is a normed vector space over the number field   or   (the real numbers or the complex numbers), with a norm   Consider its dual normed space   that consists of all continuous linear functionals   and is equipped with the dual norm   defined by

 

The dual   is a normed space (a Banach space to be precise), and its dual normed space   is called bidual space for   The bidual consists of all continuous linear functionals   and is equipped with the norm   dual to   Each vector   generates a scalar function   by the formula:

 
and   is a continuous linear functional on   that is,   One obtains in this way a map
 
called evaluation map, that is linear. It follows from the Hahn–Banach theorem that   is injective and preserves norms:
 
that is,   maps   isometrically onto its image   in   Furthermore, the image   is closed in   but it need not be equal to  

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

  1. the evaluation map   is surjective,
  2. the evaluation map   is an isometric isomorphism of normed spaces,
  3. the evaluation map   is an isomorphism of normed spaces.

A reflexive space   is a Banach space, since   is then isometric to the Banach space  

Remark edit

A Banach space   is reflexive if it is linearly isometric to its bidual under this canonical embedding   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   has codimension one in its bidual. [2] A Banach space   is called quasi-reflexive (of order  ) if the quotient   has finite dimension  

Examples edit

  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   from the definition is bijective, by the rank–nullity theorem.
  2. The Banach space   of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that   and   are not reflexive, because   is isomorphic to the dual of   and   is isomorphic to the dual of  
  3. All Hilbert spaces are reflexive, as are the Lp spaces   for   More generally: all uniformly convex Banach spaces are reflexive according to the Milman–Pettis theorem. The   and   spaces are not reflexive (unless they are finite dimensional, which happens for example when   is a measure on a finite set). Likewise, the Banach space   of continuous functions on   is not reflexive.
  4. The spaces   of operators in the Schatten class on a Hilbert space   are uniformly convex, hence reflexive, when   When the dimension of   is infinite, then   (the trace class) is not reflexive, because it contains a subspace isomorphic to   and   (the bounded linear operators on  ) is not reflexive, because it contains a subspace isomorphic to   In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of  

Properties edit

Since every finite-dimensional normed space is a reflexive Banach space, only infinite-dimensional spaces can be non-reflexive.

If a Banach space   is isomorphic to a reflexive Banach space   then   is reflexive.[3]

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.[4]

Let   be a Banach space. The following are equivalent.

  1. The space   is reflexive.
  2. The continuous dual of   is reflexive.[5]
  3. The closed unit ball of   is compact in the weak topology. (This is known as Kakutani's Theorem.)[6]
  4. Every bounded sequence in   has a weakly convergent subsequence.[7]
  5. The statement of Riesz's lemma holds when the real number[note 1] is exactly  [8] Explicitly, for every closed proper vector subspace   of   there exists some vector   of unit norm   such that   for all  
    • Using   to denote the distance between the vector   and the set   this can be restated in simpler language as:   is reflexive if and only if for every closed proper vector subspace   there is some vector   on the unit sphere of   that is always at least a distance of   away from the subspace.
    • For example, if the reflexive Banach space   is endowed with the usual Euclidean norm and   is the   plane then the points   satisfy the conclusion   If   is instead the  -axis then every point belonging to the unit circle in the   plane satisfies the conclusion.
  6. Every continuous linear functional on   attains its supremum on the closed unit ball in  [9] (James' theorem)

Since norm-closed convex subsets in a Banach space are weakly closed,[10] it follows from the third property that closed bounded convex subsets of a reflexive space   are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of   the intersection is non-empty. As a consequence, every continuous convex function   on a closed convex subset   of   such that the set

 
is non-empty and bounded for some real number   attains its minimum value on  

The promised geometric property of reflexive Banach spaces is the following: if   is a closed non-empty convex subset of the reflexive space   then for every   there exists a   such that   minimizes the distance between   and points of   This follows from the preceding result for convex functions, applied to  Note that while the minimal distance between   and   is uniquely defined by   the point   is not. The closest point   is unique when   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   separability of the continuous dual   implies separability of  [11]

Super-reflexive space edit

Informally, a super-reflexive Banach space   has the following property: given an arbitrary Banach space   if all finite-dimensional subspaces of   have a very similar copy sitting somewhere in   then   must be reflexive. By this definition, the space   itself must be reflexive. As an elementary example, every Banach space   whose two dimensional subspaces are isometric to subspaces of   satisfies the parallelogram law, hence[12]   is a Hilbert space, therefore   is reflexive. So   is super-reflexive.

The formal definition does not use isometries, but almost isometries. A Banach space   is finitely representable[13] in a Banach space   if for every finite-dimensional subspace   of   and every   there is a subspace   of   such that the multiplicative Banach–Mazur distance between   and   satisfies

 

A Banach space finitely representable in   is a Hilbert space. Every Banach space is finitely representable in   The Lp space   is finitely representable in  

A Banach space   is super-reflexive if all Banach spaces   finitely representable in   are reflexive, or, in other words, if no non-reflexive space   is finitely representable in   The notion of ultraproduct of a family of Banach spaces[14] allows for a concise definition: the Banach space   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.[13]

Finite trees in Banach spaces edit

One of James' characterizations of super-reflexivity uses the growth of separated trees.[15] The description of a vectorial binary tree begins with a rooted binary tree labeled by vectors: a tree of height   in a Banach space   is a family of   vectors of   that can be organized in successive levels, starting with level 0 that consists of a single vector   the root of the tree, followed, for   by a family of  2 vectors forming level  

 
that are the children of vertices of level   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:
 

Given a positive real number   the tree is said to be  -separated if for every internal vertex, the two children are  -separated in the given space norm:

 

Theorem.[15] The Banach space   is super-reflexive if and only if for every   there is a number   such that every  -separated tree contained in the unit ball of   has height less than  

Uniformly convex spaces are super-reflexive.[15] Let   be uniformly convex, with modulus of convexity   and let   be a real number in   By the properties of the modulus of convexity, a  -separated tree of height   contained in the unit ball, must have all points of level   contained in the ball of radius   By induction, it follows that all points of level   are contained in the ball of radius

 

If the height   was so large that

 
then the two points   of the first level could not be  -separated, contrary to the assumption. This gives the required bound   function of   only.

Using the tree-characterization, Enflo proved[16] 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[17] that a super-reflexive space   admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant   and some real number  

 

Reflexive locally convex spaces edit

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

Let   be a topological vector space over a number field   (of real numbers   or complex numbers  ). Consider its strong dual space   which consists of all continuous linear functionals   and is equipped with the strong topology   that is,, the topology of uniform convergence on bounded subsets in   The space   is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space   which is called the strong bidual space for   It consists of all continuous linear functionals   and is equipped with the strong topology   Each vector   generates a map   by the following formula:

 
This is a continuous linear functional on   that is,,   This induces a map called the evaluation map:
 
This map is linear. If   is locally convex, from the Hahn–Banach theorem it follows that   is injective and open (that is, for each neighbourhood of zero   in   there is a neighbourhood of zero   in   such that  ). But it can be non-surjective and/or discontinuous.

A locally convex space   is called

  • semi-reflexive if the evaluation map   is surjective (hence bijective),
  • reflexive if the evaluation map   is surjective and continuous (in this case   is an isomorphism of topological vector spaces[18]).

Theorem[19] — A locally convex Hausdorff space   is semi-reflexive if and only if   with the  -topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of   are weakly compact).

Theorem[20][21] — A locally convex space   is reflexive if and only if it is semi-reflexive and barreled.

Theorem[22] — The strong dual of a semireflexive space is barrelled.

Theorem[23] — If   is a Hausdorff locally convex space then the canonical injection from   into its bidual is a topological embedding if and only if   is infrabarreled.

Semireflexive spaces edit

Characterizations edit

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

  1.   is semireflexive;
  2. The weak topology on   had the Heine-Borel property (that is, for the weak topology   every closed and bounded subset of   is weakly compact).[1]
  3. If linear form on   that continuous when   has the strong dual topology, then it is continuous when   has the weak topology;[24]
  4.   is barreled;[24]
  5.   with the weak topology   is quasi-complete.[24]

Characterizations of reflexive spaces edit

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

  1.   is reflexive;
  2.   is semireflexive and infrabarreled;[23]
  3.   is semireflexive and barreled;
  4.   is barreled and the weak topology on   had the Heine-Borel property (that is, for the weak topology   every closed and bounded subset of   is weakly compact).[1]
  5.   is semireflexive and quasibarrelled.[25]

If   is a normed space then the following are equivalent:

  1.   is reflexive;
  2. The closed unit ball is compact when   has the weak topology  [26]
  3.   is a Banach space and   is reflexive.[27]
  4. Every sequence   with   for all   of nonempty closed bounded convex subsets of   has nonempty intersection.[28]

Theorem[29] — 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   is reflexive if and only if every continuous linear functional on   attains its supremum on the closed unit ball in  

Sufficient conditions edit

Normed spaces

A normed space that is semireflexive is a reflexive Banach space.[30] A closed vector subspace of a reflexive Banach space is reflexive.[23]

Let   be a Banach space and   a closed vector subspace of   If two of   and   are reflexive then they all are.[23] This is why reflexivity is referred to as a three-space property.[23]

Topological vector spaces

If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.[1]

The strong dual of a reflexive space is reflexive.[31]Every Montel space is reflexive.[26] And the strong dual of a Montel space is a Montel space (and thus is reflexive).[26]

Properties edit

A locally convex Hausdorff reflexive space is barrelled. If   is a normed space then   is an isometry onto a closed subspace of  [30] This isometry can be expressed by:

 

Suppose that   is a normed space and   is its bidual equipped with the bidual norm. Then the unit ball of     is dense in the unit ball   of   for the weak topology  [30]

Examples edit

  1. Every finite-dimensional Hausdorff topological vector space is reflexive, because   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   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   its dual normed space   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:
    1.   is a reflexive normed space (that is,   is an isomorphism of normed spaces),
    2.   is a reflexive locally convex space (that is,   is an isomorphism of topological vector spaces[18]),
    3.   is a semi-reflexive locally convex space (that is,   is surjective).
  3. A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let   be an infinite dimensional reflexive Banach space, and let   be the topological vector space   that is, the vector space   equipped with the weak topology. Then the continuous dual of   and   are the same set of functionals, and bounded subsets of   (that is, weakly bounded subsets of  ) are norm-bounded, hence the Banach space   is the strong dual of   Since   is reflexive, the continuous dual of   is equal to the image   of   under the canonical embedding   but the topology on   (the weak topology of  ) is not the strong topology   that is equal to the norm topology of  
  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   of smooth functions on arbitrary (real) smooth manifold   and its strong dual space   of distributions with compact support on  
    • the space   of smooth functions with compact support on arbitrary (real) smooth manifold   and its strong dual space   of distributions on  
    • the space   of holomorphic functions on arbitrary complex manifold   and its strong dual space   of analytic functionals on  
    • the Schwartz space   on   and its strong dual space   of tempered distributions on  

Counter-examples edit

  • There exists a non-reflexive locally convex TVS whose strong dual is reflexive.[33]

Other types of reflexivity edit

A stereotype space, or polar reflexive space, is defined as a topological vector space (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space   More precisely, a TVS   is called polar reflexive[34] or stereotype if the evaluation map into the second dual space

 
is an isomorphism of topological vector spaces.[18] Here the stereotype dual space   is defined as the space of continuous linear functionals   endowed with the topology of uniform convergence on totally bounded sets in   (and the stereotype second dual space   is the space dual to   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   in the definition of dual space   by other classes of subsets, for example, by the class of compact subsets in   – the spaces defined by the corresponding reflexivity condition are called reflective,[35][36] 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 edit

  • Grothendieck space
    • 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 – operator algebra that has enough invariant subspaces to characterize it

References edit

Notes edit

  1. ^ The statement of Riesz's lemma involves only one real number, which is denoted by   in the article on Riesz's lemma. The lemma always holds for all real   But for a Banach space, the lemma holds for all   if and only if the space is reflexive.

Citations edit

  1. ^ a b c d e Trèves 2006, pp. 372–374.
  2. ^ Robert C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998.
  3. ^ Proposition 1.11.8 in Megginson (1998, p. 99).
  4. ^ Megginson (1998, pp. 104–105).
  5. ^ Corollary 1.11.17, p. 104 in Megginson (1998).
  6. ^ Conway 1985, Theorem V.4.2, p. 135.
  7. ^ Since weak compactness and weak sequential compactness coincide by the Eberlein–Šmulian theorem.
  8. ^ Diestel 1984, p. 6.
  9. ^ Theorem 1.13.11 in Megginson (1998, p. 125).
  10. ^ Theorem 2.5.16 in Megginson (1998, p. 216).
  11. ^ Theorem 1.12.11 and Corollary 1.12.12 in Megginson (1998, pp. 112–113).
  12. ^ see this characterization of Hilbert space among Banach spaces
  13. ^ a b James, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896–904.
  14. ^ 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.
  15. ^ a b c see James (1972).
  16. ^ Enflo, Per (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics. 13: 281–288. doi:10.1007/BF02762802.
  17. ^ Pisier, Gilles (1975). "Martingales with values in uniformly convex spaces". Israel Journal of Mathematics. 20: 326–350. doi:10.1007/BF02760337.
  18. ^ a b c An isomorphism of topological vector spaces is a linear and a homeomorphic map  
  19. ^ Edwards 1965, 8.4.2.
  20. ^ Schaefer 1966, 5.6, 5.5.
  21. ^ Edwards 1965, 8.4.5.
  22. ^ Edwards 1965, 8.4.3.
  23. ^ a b c d e Narici & Beckenstein 2011, pp. 488–491.
  24. ^ a b c Schaefer & Wolff 1999, p. 144.
  25. ^ Khaleelulla 1982, pp. 32–63.
  26. ^ a b c Trèves 2006, p. 376.
  27. ^ Trèves 2006, p. 377.
  28. ^ Bernardes 2012.
  29. ^ Narici & Beckenstein 2011, pp. 212.
  30. ^ a b c Trèves 2006, p. 375.
  31. ^ Schaefer & Wolff 1999, p. 145.
  32. ^ Edwards 1965, 8.4.7.
  33. ^ Schaefer & Wolff 1999, pp. 190–202.
  34. ^ Köthe, Gottfried (1983). Topological Vector Spaces I. Springer Grundlehren der mathematischen Wissenschaften. Springer. ISBN 978-3-642-64988-2.
  35. ^ 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 (1–2): 75–89. doi:10.1016/s0166-8641(01)00111-0.
  36. ^ Akbarov, S. S.; Shavgulidze, E. T. (2003). "On two classes of spaces reflexive in the sense of Pontryagin". Mat. Sbornik. 194 (10): 3–26.

General references edit

  • Bernardes, Nilson C. Jr. (2012), On nested sequences of convex sets in Banach spaces, vol. 389, Journal of Mathematical Analysis and Applications, pp. 558–561 .
  • Conway, John B. (1985). A Course in Functional Analysis. Springer.
  • Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781.
  • Edwards, R. E. (1965). Functional analysis. Theory and applications. New York: Holt, Rinehart and Winston. ISBN 0030505356.
  • James, Robert C. (1972), Some self-dual properties of normed linear spaces. Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), Ann. of Math. Studies, vol. 69, Princeton, NJ: Princeton Univ. Press, pp. 159–175.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Kolmogorov, A. N.; Fomin, S. V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press.
  • Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  • Schaefer, Helmut H. (1966). Topological vector spaces. New York: The Macmillan Company.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.