Skip to main content
SpringerLink
Log in

Nonstandard hulls of Banach spaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The main theme of this paper is the relationship between a Banach spaceE and its nonstandard hullsÊ (including ultrapowers ofE). Emphasis is placed on the ways in which the general structure ofÊ is determined by the approximate shape and arrangement of the finite dimensional subspaces ofE.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. C. Chang and H. J. Keisler,Model Theory, North-Holland, Amsterdam, 1973.

    MATH  Google Scholar 

  2. D. Cozart and L. C. Moore, Jr.The nonstandard hull of a normed Riesz space, Duke Math. J.41 (1974), 263–275.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Dacunha-Castelle and J.-L. Krivine,Applications des ultraproduits à l'étude des espaces et des algèbres de Banach, Studia Math.41 (1972), 315–334.

    MATH  MathSciNet  Google Scholar 

  4. L. Gillman and M. Jerison,Rings of Continuous Functions, D. van Nostrand Co., Inc., New York, 1960.

    MATH  Google Scholar 

  5. C. Ward Henson,The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces, J. Symb. Logic39 (1974), 717–731.

    Article  MathSciNet  Google Scholar 

  6. C. Ward Henson,When do two Banach spaces have isometrically isomorphic nonstandard hulls?, Israel J. Math.22 (1975), 57–67.

    MATH  MathSciNet  Google Scholar 

  7. C. Ward Henson,Model theory of Banach spaces (in preparation).

  8. C. Ward Henson, C. G. Jockusch, Jr., L. A. Rubel and G. Takeuti,First-order topology, (to appear in Dissertationes Mathematicae).

  9. C. Ward Henson and L. C. Moore, Jr.,The nonstandard theory of topological vector spaces, Trans. Amer. Math. Soc.172 (1972), 405–435.

    Article  MathSciNet  Google Scholar 

  10. C. Ward Henson and L. C. Moore, Jr.,Nonstandard hulls of the classical Banach spaces, Duke Math. J.41 (1974), 227–284.

    Article  MathSciNet  Google Scholar 

  11. C. Ward Henson and L. C. Moore, Jr.,Subspaces of the nonstandard hull of a normed space, Trans. Amer. Math. Soc.197 (1974), 131–143.

    Article  MATH  MathSciNet  Google Scholar 

  12. J.-L. Krivine,Théorie des modèles et espaces L p, C. R. Acad. Sci. Paris Ser. A275 (1972), 1207–1210.

    MATH  MathSciNet  Google Scholar 

  13. J.-L. Krivine,Langages à valeurs réelles et applications, Fund. Math.81 (1974), 213–253.

    MATH  MathSciNet  Google Scholar 

  14. H. E. Lacey,The Isometric Theory of Classical Banach Spaces, Springer-Verlag, New York, 1974.

    MATH  Google Scholar 

  15. P. Loeb,Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc.211 (1975), 113–122.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. A. J. Luxemburg,A general theory of monads, inApplications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston, New York, 1969, pp. 18–85.

    Google Scholar 

  17. A Robinson,Nonstandard Analysis North-Holland, Amsterdam, 1966.

    Google Scholar 

  18. A. Robinson and E. Zakon,A set-theoretical characterization of enlargements, inApplications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston, New York, 1969, pp. 109–122.

    Google Scholar 

  19. J. R. Shoenfield,Mathematical Logic, Addison-Wesley, Reading, Mass., 1967.

    MATH  Google Scholar 

  20. J. Stern,Sur certaines classes d'espaces de Banach caractérisées par des formules, C. R. Acad. Sci. Paris Ser. A278 (1974), 525–528.

    MATH  MathSciNet  Google Scholar 

  21. J. Stern,Some applications of model theory in Banach space theory, Ann. Math. Logic9 (1976), 49–121.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Stern,The problem of envelopes for Banach spaces, (to appear).

  23. J. Stern,Ultrapowers and local properties of Banach spaces, (to appear).

  24. A. TarskiArithmetical classes and types of Boolean algebras, (preliminary report), Bull. Amer. Math. Soc.55 (1949), 64.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. New Mexico State University and University of Illinois, Urbana/Champaign, Ill., USA

    C. Ward Henson

Authors
  1. C. Ward Henson
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was partially supported by a grant from the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henson, C.W. Nonstandard hulls of Banach spaces. Israel J. Math. 25, 108–144 (1976). https://doi.org/10.1007/BF02756565

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02756565

Keywords

Search

Navigation