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Approximation of lipschitz functions by Δ-convex functions in banach spaces

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Abstract

In this paper we give some result about the approximation of a Lipschitz function on a Banach space by means of Δ-convex functions. In particular, we prove that the density of Δ-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of Δ-convex functions.

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References

  1. E. Asplund,Averaged norms, Israel Journal of Mathematics5 (1967), 227–233.

    MATH  MathSciNet  Google Scholar 

  2. R. Deville, V. Fonf and P. Hájek,Analytic and C k approximations of norms in separable Banach spaces, Studia Mathematica120 (1996), 61–74.

    MATH  MathSciNet  Google Scholar 

  3. R. Deville, V. Fonf and P. Hájek,Analytic and polyhedral approximation of convex bodies in separable polyhedral spaces, to appear.

  4. R. Deville, G. Godefroy and V. Zizler,Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Boston, 1993.

    MATH  Google Scholar 

  5. G. A. Edgar,Measurability in a Banach space, Indiana University Mathematics Journal26 (1977), 663–677.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Hoffman-Jørgensen,On the Modulus of Smoothness and the G * -Conditions in B-spaces, Preprint series, Aarhus Universitet, Matematisk Institute, 1974.

  7. R. C. James,Some self-dual properties of normed linear spaces, inSymposium on Infinite-Dimensional Topology, Annals of Mathematics Studies69, Princeton University Press, 1972, pp. 159–175.

  8. J. Kurzweil,On approximation in real Banach spaces, Studia Mathematica14 (1954), 214–231.

    MathSciNet  Google Scholar 

  9. G. Lancien,On uniformly convex and uniformly Kadec-Klee renormings, Serdica Mathematics Journal21 (1995), 1–18.

    MATH  MathSciNet  Google Scholar 

  10. J. M. Lasry and P. L. Lions,A remark on regularization in Hilbert spaces, Isreal Journal of Mathematics55 (1986), 257–266.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Odell and T. Schlumprecht,The distortion problem, Acta Mathematica173 (1994), 259–281.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. R. Phelps,Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  13. G. Pisier,Martingales with values in uniformly convex spaces, Israel Journal of Mathematics20 (1975), 326–350.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Poliquin, J. Vanderwerff and V. Zizler,Convex composite representation of lower semicontinuous functions and renormings, Comptes Rendus de l’Académie des Sciences, Paris, Série I317 (1993), 545–549.

    MATH  MathSciNet  Google Scholar 

  15. C. Stegall,The Radon-Nikodym property in conjugate Banach spaces, Transactions of the American Mathematical Society206 (1975), 213–223.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Stern,Propriétés locales et ultrapuissances d’espaces de Banach. Exposés 7 et 8, Séminaire Maurey-Schwartz, Centre Math., École Polytech, Paris, 1974–1975.

    Google Scholar 

  17. T. Strömberg,On regularization in Banach spaces, Arkiv för Matematik34 (1996), 383–406.

    Article  MATH  Google Scholar 

  18. M. Talagrand,Comparaison des boréliens d’un espace de Banach pour les topologies fortes et faibles, Indiana University Mathematics Journal27 (1978), 1001–1004.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Equipe d’Analyse - Boite 186, Tour 46-00, 4-ième étage, Université Paris 6, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Manuel Cepedello Boiso

  2. Departmento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, 41080, Sevilla, Spain

    Manuel Cepedello Boiso

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  1. Manuel Cepedello Boiso
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Correspondence to Manuel Cepedello Boiso.

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Boiso, M.C. Approximation of lipschitz functions by Δ-convex functions in banach spaces. Isr. J. Math. 106, 269–284 (1998). https://doi.org/10.1007/BF02773472

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  • DOI: https://doi.org/10.1007/BF02773472

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