Skip to main content
SpringerLink
Log in

The problem of envelopes for Banach spaces

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

Abstract

LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol 2, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ1 (assuming the continuum hypothesis).

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, 1973.

  2. D. Dacunha-Castelle et J. L. Krivine,Application des ultraproduits à l’étude des espaces et des algèbres de Banach, Studia Math.41 (1972), 315–344.

    MATH  MathSciNet  Google Scholar 

  3. J. Lindenstrauss and A. Pelczynski,Absolutely summing operators in ℒ p p and their applications, Studia Math.29 (1968), 275–326.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. J. Stern, Exposés VII–VIII inSéminaire Maurey-Schwartz, Ecole Polytechnique, 1974–1975.

Download references

Author information

Authors and Affiliations

  1. Université De Paris VII, Paris, France

    Jacques Stern

Authors
  1. Jacques Stern
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stern, J. The problem of envelopes for Banach spaces. Israel J. Math. 24, 1–15 (1976). https://doi.org/10.1007/BF02761425

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761425

Keywords

Search

Navigation