Semi-reflexive space

(Redirected from Semireflexive space)

In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of X) is bijective. If this map is also an isomorphism of TVSs then it is called reflexive.

Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable.

Definition and notation

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Brief definition

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

For any   let   be defined by  , where   is called the evaluation map at x; since   is necessarily continuous, it follows that  . Since   separates points on X, the map   defined by   is injective where this map is called the evaluation map or the canonical map. This map was introduced by Hans Hahn in 1927.[2]

We call X semireflexive if   is bijective (or equivalently, surjective) and we call X reflexive if in addition   is an isomorphism of TVSs.[1] If X is a normed space then J is a TVS-embedding as well as an isometry onto its range; furthermore, by Goldstine's theorem (proved in 1938), the range of J is a dense subset of the bidual  .[2] A normable space is reflexive if and only if it is semi-reflexive. A Banach space is reflexive if and only if its closed unit ball is  -compact.[2]

Detailed definition

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Let X 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 X. 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 X. 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,  . One obtains a map called the evaluation map or the canonical injection:

 

which is a linear map. If X is locally convex, from the Hahn–Banach theorem it follows that J is injective and open (that is, for each neighbourhood of zero   in X there is a neighbourhood of zero V 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); it is called reflexive if the evaluation map   is surjective and continuous, in which case J will be an isomorphism of TVSs).

Characterizations of semi-reflexive spaces

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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 (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;[3]
  4.   is barrelled, where the   indicates the Mackey topology on  ;[3]
  5. X weak the weak topology   is quasi-complete.[3]

Theorem[4] — 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).

Sufficient conditions

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Every semi-Montel space is semi-reflexive and every Montel space is reflexive.

Properties

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If   is a Hausdorff locally convex space then the canonical injection from   into its bidual is a topological embedding if and only if   is infrabarrelled.[5]

The strong dual of a semireflexive space is barrelled. Every semi-reflexive space is quasi-complete.[3] Every semi-reflexive normed space is a reflexive Banach space.[6] The strong dual of a semireflexive space is barrelled.[7]

Reflexive spaces

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

  1. X is reflexive;
  2. X is semireflexive and barrelled;
  3. X is barrelled and the weak topology on X had the Heine-Borel property (which means that for the weak topology  , every closed and bounded subset of   is weakly compact).[1]
  4. X is semireflexive and quasibarrelled.[8]

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  .[9]
  3. X is a Banach space and   is reflexive.[10]

Examples

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Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive.[11] If   is a dense proper vector subspace of a reflexive Banach space then   is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space.[11] There exists a semi-reflexive countably barrelled space that is not barrelled.[11]

See also

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Citations

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  1. ^ a b c d Trèves 2006, pp. 372–374.
  2. ^ a b c Narici & Beckenstein 2011, pp. 225–273.
  3. ^ a b c d Schaefer & Wolff 1999, p. 144.
  4. ^ Edwards 1965, 8.4.2.
  5. ^ Narici & Beckenstein 2011, pp. 488–491.
  6. ^ Schaefer & Wolff 1999, p. 145.
  7. ^ Edwards 1965, 8.4.3.
  8. ^ Khaleelulla 1982, pp. 32–63.
  9. ^ Trèves 2006, p. 376.
  10. ^ Trèves 2006, p. 377.
  11. ^ a b c Khaleelulla 1982, pp. 28–63.

Bibliography

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  • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
  • Edwards, R. E. (1965). Functional analysis. Theory and applications. New York: Holt, Rinehart and Winston. ISBN 0030505356.
  • John B. Conway, A Course in Functional Analysis, Springer, 1985.
  • 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.
  • 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.
  • Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.