Israel Journal of Mathematics

, Volume 44, Issue 3, pp 262–276

Norms with locally Lipschitzian derivatives

  • M. Fabian
  • J. H. M. Whitfield
  • V. Zizler
Article

Abstract

If a separable Banach spaceX admits a real valued function ф with bounded nonempty support, φ 艂 is locally Lipschitzian and if no subspace ofX is isomorphic toco, thenX admits an equivalent twice Gateaux differentiable norm whose first Frechet differential is Lipschitzian on the unit sphere ofX.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Bessaga and A. Pelczynski,On bases and unconditional convergence of series in Banach spaces, Studia Math.17 (1958), 151–164.MATHMathSciNetGoogle Scholar
  2. 2.
    R. Bonic and J. Frampton,Smooth functions on Banach manifolds, J. Math. Mech.,15 (1966), 877–898.MATHMathSciNetGoogle Scholar
  3. 3.
    M. M. Day,Uniform convexity III, Bull. Am. Math. Soc.49 (1943), 745–750.MATHGoogle Scholar
  4. 4.
    M. M. Day,Strict convexity and smoothness, Trans. Am. Math. Soc.78 (1955), 516–528.MATHCrossRefGoogle Scholar
  5. 5.
    J. Diestel,Geometry of Banach Spaces, Selected Topics, Lecture Notes in Math., No. 485, Springer-Verlag, 1975.Google Scholar
  6. 6.
    P. Enflo,Banach spaces which can be given an equivalent uniformly convex norm, Isr. J. Math.13 (1972), 281–288.CrossRefMathSciNetGoogle Scholar
  7. 7.
    T. Figiel,Uniformly convex norms in spaces with unconditional bases, Seminaire Maurey-Schwartz, Ecole Polytechnique, Paris, 1974–1975.Google Scholar
  8. 8.
    T. Figiel,On the moduli of convexity and smoothness, Studia Math.56 (1976), 121–155.MATHMathSciNetGoogle Scholar
  9. 9.
    T. Figiel and G. Pisier,Series aléatoires dans les espaces uniformément convèxes ou uniformément lissés, C. R. Acad. Sci. Paris, Ser. A279 (1974), 611–614.MATHMathSciNetGoogle Scholar
  10. 10.
    J. Hoffman-Jorgensen,Sums of independent Banach space valued random variables, Aarhus Universitat Preprint Series No. 15 (1972–1973).Google Scholar
  11. 11.
    R. C. James,Superreflexive Banach spaces, Can. J. Math.24 (1972), 896–904.MATHGoogle Scholar
  12. 12.
    R. C. James,Nonreflexive spaces of type 2, Isr. J. Math.,30 (1978), 1–13.MATHGoogle Scholar
  13. 13.
    J. Kurzweil,On approximation in real Banach spaces, Studia Math.14 (1954), 213–231.MathSciNetGoogle Scholar
  14. 14.
    S. Kwapień,Isomorphic characterization on inner product spaces by orthogonal series with vector valued coefficients, Studia Math.44 (1972), 583–595.MathSciNetGoogle Scholar
  15. 15.
    E. B. Leach and J. H. M. Whitfield,Differentiable functions and rough norms on Banach spaces, Proc. Am. Math. Soc.33 (1972), 120–126.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. Lindenstrauss,On operators which attain their norms, Isr. J. Math.1 (1963), 139–148.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, 1977.Google Scholar
  18. 18.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Function Spaces, Springer-Verlag, 1978.Google Scholar
  19. 19.
    B. M. Makarov,One characteristic of Hilbert space, Mat. Zamet.26 (1979), 739–746 (Russian).MATHGoogle Scholar
  20. 20.
    V. Z. Meshkov,On smooth functions on James space, Vesnik Moskov. Univ.4 (1974), 9–13 (Russian).Google Scholar
  21. 21.
    V. Z. Meshkov,Smoothness properties in Banach spaces, Studia Math.63 (1978), 111–123.MATHMathSciNetGoogle Scholar
  22. 22.
    J. Pechanec, J. H. M. Whitfield and V. Zizler,Norms locally dependent on finitely many coordinates, Annais de Acad. Brasil, to appear.Google Scholar
  23. 23.
    M. M. Rao,Characterization of Hilbert spaces by smoothness, Nederl. Acad. v. W. A.71 (1967), 132–135.Google Scholar
  24. 24.
    K. Sundaresan,Smooth Banach spaces, Bull. Am. Math. Soc.72 (1966), 520–521.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1983

Authors and Affiliations

  • M. Fabian
    • 1
    • 2
    • 3
  • J. H. M. Whitfield
    • 1
    • 2
    • 3
  • V. Zizler
    • 1
    • 2
    • 3
  1. 1.Prague 6Czechoslovakia
  2. 2.Department of Mathematical SciencesLakehead UniversityThunder BayCanada
  3. 3.Prague 4Czechoslovakia

Personalised recommendations