×

zbMATH — the first resource for mathematics

Approximation of Lipschitz functions by \(\Delta\)-convex functions in Banach spaces. (English) Zbl 0920.46010
Summary: We give some results about the approximation of a Lipschitz function on a Banach space by means of \(\Delta\)-convex functions. In particular, we prove that the density of \(\Delta\)-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 \(\Delta\)-convex functions.

MSC:
46B20 Geometry and structure of normed linear spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
41A30 Approximation by other special function classes
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] E. Asplund,Averaged norms, Israel Journal of Mathematics5 (1967), 227–233. · Zbl 0153.44301
[2] R. Deville, V. Fonf and P. Hájek,Analytic and C k approximations of norms in separable Banach spaces, Studia Mathematica120 (1996), 61–74. · Zbl 0857.46008
[3] R. Deville, V. Fonf and P. Hájek,Analytic and polyhedral approximation of convex bodies in separable polyhedral spaces, to appear. · Zbl 0920.46011
[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. · Zbl 0782.46019
[5] G. A. Edgar,Measurability in a Banach space, Indiana University Mathematics Journal26 (1977), 663–677. · Zbl 0361.46017
[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.
[9] G. Lancien,On uniformly convex and uniformly Kadec-Klee renormings, Serdica Mathematics Journal21 (1995), 1–18. · Zbl 0837.46011
[10] J. M. Lasry and P. L. Lions,A remark on regularization in Hilbert spaces, Isreal Journal of Mathematics55 (1986), 257–266. · Zbl 0631.49018
[11] E. Odell and T. Schlumprecht,The distortion problem, Acta Mathematica173 (1994), 259–281. · Zbl 0828.46005
[12] R. R. Phelps,Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin, 1993. · Zbl 0921.46039
[13] G. Pisier,Martingales with values in uniformly convex spaces, Israel Journal of Mathematics20 (1975), 326–350. · Zbl 0344.46030
[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. · Zbl 0801.46007
[15] C. Stegall,The Radon-Nikodym property in conjugate Banach spaces, Transactions of the American Mathematical Society206 (1975), 213–223. · Zbl 0318.46056
[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.
[17] T. Strömberg,On regularization in Banach spaces, Arkiv för Matematik34 (1996), 383–406. · Zbl 0873.49028
[18] M. Talagrand,Comparaison des boréliens d’un espace de Banach pour les topologies fortes et faibles, Indiana University Mathematics Journal27 (1978), 1001–1004. · Zbl 0396.28007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.