Skip to main content
SpringerLink
Log in

Boundaries for Banach spaces determine weak compactness

  • Published:
Inventiones mathematicae Aims and scope

Abstract

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.

The main theorem of this paper states the equivalence of countable and sequential compactness of norm bounded sets with respect to an appropriate topology. This result contains, as a special case, the positive answer to the boundary problem and it carries James’ sup-characterization as a corollary.

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. Behrends, E.: New proofs of Rosenthal’s l 1-theorem and the Josefson-Nissenzweig theorem. Bull. Pol. Acad. Sci. Math. 43, 283–295 (1996)

    MathSciNet  Google Scholar 

  2. Bourgain, J.: La propriété de Radon-Nikodym. In: Publications Mathématiques de l’Université Pierre et Marie Curie Paris VI, Cours de troisième cycle, n. 36 (1979)

  3. Bourgain, J., Fremlin, D.H., Talagrand, M.: Pointwise compact sets of Baire-measurable functions. Am. J. Math. 100, 845–886 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bourgain, J., Talagrand, M.: Compacité extrémale. Proc. Am. Math. Soc. 80, 68–70 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cascales, B., Godefroy, G.: Angelicity and the boundary problem. Mathematika 45, 105–112 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cascales, B., Manjabacas, G., Vera, G.: A Krein-Šmulian type result in Banach spaces. Quart. J. Math. Oxford (2) 48, 161–167 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cascales, B., Shvydkoy, R.: On the Krein-Šmulian theorem for weaker topologies. Ill. J. Math. 47, 957–976 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin/Heidelberg/New York (1984)

    Google Scholar 

  9. Dowling, P.N., Johnson, W.B., Lennard, C.J., Turett, B.: The optimality of James’s distortion theorems. Proc. Am. Math. Soc. 125, 167–174 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  11. Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J., Zizler, V.: Functional analysis and infinite-dimensional geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8. Springer, Berlin (2001)

    MATH  Google Scholar 

  12. Floret, K.: Weakly Compact Sets. Lecture Notes in Math., vol. 801. Springer, Berlin/Heidelberg/New York (1980)

    MATH  Google Scholar 

  13. Fonf, V.P., Lindenstrauss, J.: Boundaries and generation of convex sets. Isr. J. Math. 136, 157–172 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fonf, V.P., Lindenstrauss, J., Phelps, R.R.: Infinite dimensional convexity. Chap. 15 of [21]

  15. Godefroy, G.: Boundaries of a convex set and interpolation sets. Math. Ann. 277, 173–184 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. Godefroy, G.: Some applications of Simons’s inequality. Serdica Math. J. 26, 59–78 (2000)

    MATH  MathSciNet  Google Scholar 

  17. Granero, A.S., Hernández, J.M.: On James boundaries in dual Banach spaces. Preprint (2010)

  18. Hagler, J., Johnson, W.B.: On Banach spaces whose dual balls are not weak∗ sequentially compact. Isr. J. Math. 28, 325–330 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. James, R.C.: Reflexivity and the supremum of linear functionals (1957)

  20. James, R.C.: Weak compactness and reflexivity. Isr. J. Math. 2, 101–119 (1964)

    Article  MATH  Google Scholar 

  21. Johnson, W.B., Lindenstrauss, J.: Handbook of the Geometry of Banach Spaces, Vols. 1 and 2. North Holland, Amsterdam (2001) 2003

    Google Scholar 

  22. Kalenda, O.F.K.: (I)-envelopes of closed convex sets in Banach spaces. Isr. J. Math. 162, 157–181 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kalenda, O.F.K., Spurný, J.: Boundaries of compact convex sets and fragmentability. J. Funct. Anal. 256, 865–880 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Khurana, S.S.: Pointwise compactness on extreme points. Proc. Am. Math. Soc. 83, 347–348 (1981)

    MATH  MathSciNet  Google Scholar 

  25. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I and II. Springer, Berlin/Heidelberg/New York (1977) 1979

    Google Scholar 

  26. Lukeš, J., Malý, J., Netuka, I., Spurný, J.: Integral representation theory: applications to convexity, Banach spaces and potential theory. de Gruyter Studies in Mathematics, vol. 35. de Gruyter, Berlin (2009)

    Google Scholar 

  27. Morillon, M.: A new proof of James’ sup theorem. Extr. Math. 20, 261–271 (2005)

    MATH  MathSciNet  Google Scholar 

  28. Pfitzner, H.: The dual of a non-reflexive L-embedded Banach Space contains l ∞ isometrically (to appear)

  29. Rodé, G.: Superkonvexe Analysis. Arch. Math. 34, 452–462 (1980)

    Article  MATH  Google Scholar 

  30. Rodé, G.: Superkonvexität und schwache Kompaktheit. Arch. Math. 36, 62–72 (1981)

    Article  MATH  Google Scholar 

  31. Simons, S.: A convergence theorem with boundary. Pac. J. Math. 40, 703–708 (1972)

    MATH  MathSciNet  Google Scholar 

  32. Simons, S.: An eigenvector proof of Fatou’s lemma for continuous functions. Math. Intell. 17, 67–70 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. Spurný, J.: The boundary problem for L 1-preduals. Ill. J. Math. 52, 1183–1193 (2008)

    MATH  Google Scholar 

  34. De Wilde, M.: Pointwise compactness in spaces of functions and R. C. James theorem. Math. Ann. 208, 33–47 (1974)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université d’Orléans, BP 6759, 45067, Orléans Cedex 2, France

    Hermann Pfitzner

Authors
  1. Hermann Pfitzner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hermann Pfitzner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pfitzner, H. Boundaries for Banach spaces determine weak compactness. Invent. math. 182, 585–604 (2010). https://doi.org/10.1007/s00222-010-0267-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-010-0267-6

Mathematics Subject Classification (2000)

Search

Navigation