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Norms with locally Lipschitzian derivatives

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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 toc o, thenX admits an equivalent twice Gateaux differentiable norm whose first Frechet differential is Lipschitzian on the unit sphere ofX.

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References

  1. C. Bessaga and A. Pelczynski,On bases and unconditional convergence of series in Banach spaces, Studia Math.17 (1958), 151–164.

    MATH  MathSciNet  Google Scholar 

  2. R. Bonic and J. Frampton,Smooth functions on Banach manifolds, J. Math. Mech.,15 (1966), 877–898.

    MATH  MathSciNet  Google Scholar 

  3. M. M. Day,Uniform convexity III, Bull. Am. Math. Soc.49 (1943), 745–750.

    MATH  Google Scholar 

  4. M. M. Day,Strict convexity and smoothness, Trans. Am. Math. Soc.78 (1955), 516–528.

    Article  MATH  Google Scholar 

  5. J. Diestel,Geometry of Banach Spaces, Selected Topics, Lecture Notes in Math., No. 485, Springer-Verlag, 1975.

  6. P. Enflo,Banach spaces which can be given an equivalent uniformly convex norm, Isr. J. Math.13 (1972), 281–288.

    Article  MathSciNet  Google Scholar 

  7. T. Figiel,Uniformly convex norms in spaces with unconditional bases, Seminaire Maurey-Schwartz, Ecole Polytechnique, Paris, 1974–1975.

  8. T. Figiel,On the moduli of convexity and smoothness, Studia Math.56 (1976), 121–155.

    MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  10. J. Hoffman-Jorgensen,Sums of independent Banach space valued random variables, Aarhus Universitat Preprint Series No. 15 (1972–1973).

  11. R. C. James,Superreflexive Banach spaces, Can. J. Math.24 (1972), 896–904.

    MATH  Google Scholar 

  12. R. C. James,Nonreflexive spaces of type 2, Isr. J. Math.,30 (1978), 1–13.

    MATH  Google Scholar 

  13. J. Kurzweil,On approximation in real Banach spaces, Studia Math.14 (1954), 213–231.

    MathSciNet  Google Scholar 

  14. S. Kwapień,Isomorphic characterization on inner product spaces by orthogonal series with vector valued coefficients, Studia Math.44 (1972), 583–595.

    MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Lindenstrauss,On operators which attain their norms, Isr. J. Math.1 (1963), 139–148.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, 1977.

  18. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Function Spaces, Springer-Verlag, 1978.

  19. B. M. Makarov,One characteristic of Hilbert space, Mat. Zamet.26 (1979), 739–746 (Russian).

    MATH  Google Scholar 

  20. V. Z. Meshkov,On smooth functions on James space, Vesnik Moskov. Univ.4 (1974), 9–13 (Russian).

    Google Scholar 

  21. V. Z. Meshkov,Smoothness properties in Banach spaces, Studia Math.63 (1978), 111–123.

    MATH  MathSciNet  Google Scholar 

  22. J. Pechanec, J. H. M. Whitfield and V. Zizler,Norms locally dependent on finitely many coordinates, Annais de Acad. Brasil, to appear.

  23. M. M. Rao,Characterization of Hilbert spaces by smoothness, Nederl. Acad. v. W. A.71 (1967), 132–135.

    Google Scholar 

  24. K. Sundaresan,Smooth Banach spaces, Bull. Am. Math. Soc.72 (1966), 520–521.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sibelovia 49, 16200, Prague 6, Czechoslovakia

    M. Fabian, J. H. M. Whitfield & V. Zizler

  2. Department of Mathematical Sciences, Lakehead University, P7B 5E1, Thunder Bay, Ontario, Canada

    M. Fabian, J. H. M. Whitfield & V. Zizler

  3. Jihozapádní II/1084, 14100, Prague 4, Czechoslovakia

    M. Fabian, J. H. M. Whitfield & V. Zizler

Authors
  1. M. Fabian
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  2. J. H. M. Whitfield
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  3. V. Zizler
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This author's research supported in part by NSERC (Canada) Grant A7535.

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Fabian, M., Whitfield, J.H.M. & Zizler, V. Norms with locally Lipschitzian derivatives. Israel J. Math. 44, 262–276 (1983). https://doi.org/10.1007/BF02760975

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  • DOI: https://doi.org/10.1007/BF02760975

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