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Smooth bump functions and geomentry of Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

R. Deville ,
G. Godefroy  and
V. Zizler
R. Deville
Affiliation:
Université de Franche-Comté, Besaçon, France
G. Godefroy
Affiliation:
Equipe d'Analyse, Université Paris VI, F-75252-Paris, Cedex 05, France
V. Zizler
Affiliation:
Department of Mathematics, University of Alberta, 632 Central Academic Building, Edmonton, Alberta, Canada
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Abstract

Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given.

MSC classification

Secondary: 46B03: Isomorphic theory (including renorming) of Banach spaces
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 305 - 321
Copyright
Copyright © University College London 1993

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References

BF.Bonic, N. and Frampton, J.. Smooth functions on Banach manifolds. J. Math. Mech., 15 (1966), 877–898.Google Scholar
Bou.Bourgin, R. D.. Geometric aspects of convex sets with the Radon-Nikodym property. In Lecture Notes in Math., 993 (Springer-Verlag, 1983).Google Scholar
C.Collier, J. B.. The dual of a space with the Radon-Nikodym property. Pacific J. Math., 64 (1976), 103–106.CrossRefGoogle Scholar
Dl.Day, M. M.. Uniform convexity III. Bull. Amer. Math. Soc., 49 (1943), 745–750.CrossRefGoogle Scholar
D2.Day, M. M.. Strict convexity and smoothness of normed spaces. Trans. Amer. Math. Soc., 78 (1955), 516–528.Google Scholar
Dev 1.Deville, R.. Un theoreme de transfert pour la propriete des boules. Canad. Math. Bull., 30 (1987), 295–300.CrossRefGoogle Scholar
Dev 2.Deville, R.. Geometrical implications of the existence of very smooth bump functions in Banach spaces. Israel J. Math., 67 (1989), 1–22.CrossRefGoogle Scholar
DF.Deville, R. and Fabian, M.. Principes variationnels el differentiabilité d'applications deftnies sur un espace de Banach (Publ. Math. Besançon, France, 1989).Google Scholar
DGZ.Deville, R., Godefroy, G. and Zizler, V.. Smoothness and Renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics 64 (Pitman, 1993).Google Scholar
Di.Diestel, J.. Geometry of Banach spaces. Selected topics. In Lecture Notes in Math., 485 (Springer-Verlag, 1975).Google Scholar
Ed.Edgar, C. A.. A long James space. In Measure Theory, Proceedings, Lecture Notes in Math., 794 (Springer-Verlag, 1979).Google Scholar
E.Ekeland, I., Nonconvex minimization problems. Bull. Amer. Math. Soc., 1 (1979), 443–474.CrossRefGoogle Scholar
ELP.Enflo, P., Lindenstrauss, J. and Pisier, G.. On the three space problem. Math. Scand., 36 (1975), 199–210.Google Scholar
Fl.Fabian, M.. Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces. Proc London Math. Soc., 51 (1985), 113–126.Google Scholar
F2.Fabian, M.. On projectional resolution of identity on the duals of certain Banach spaces. Bull. Austr. Math. Soc., 35 (1987), 363–372.CrossRefGoogle Scholar
FPWZ.Fabian, M., Preiss, D., Whitfield, J. H. M. and Zizler, V.. Separating polynomials on Banach spaces. Quart. J. Math. Oxford (2), 40 (1989), 409–422.Google Scholar
FWZ.Fabian, M., Whitfield, J. H. M. and Zizler, V.. Norms with locally Lipschitzian derivatives. Israel J. Math., 44 (1983), 262–276.CrossRefGoogle Scholar
Fi.Finet, C.. Uniform convexity properties of norms on a superreflexive Banach space. Israel J. Math., 53 (1986), 81–92.CrossRefGoogle Scholar
GGS.Giles, J. R., Gregory, D. A. and Sims, B.. Characterization of normed spaces with Mazur's property. Bull. Austr. Math. Soc., 18 (1978), 105–123.CrossRefGoogle Scholar
GR.Griewank, A. and Rabier, P. J.. On the smoothness of convex envelopes. Trans. Amer. Math. Soc., 322 (1990), 691–709.Google Scholar
H1.Haydon, R.. A counterexample to several questions about scattered compact spaces. Bull. London Math. Soc., 22 (1990), 261–268.CrossRefGoogle Scholar
H2.Haydon, R.. Trees in renorming theory. Séminaire d'initiation à ànalyse, (1990–91).Google Scholar
H3.Haydon, R.. Infinitely differentiable norms on certain Banach spaces. C.R. Acad. Sci. Paris, Sci. Paris, Ser. I, 315 (1992), 1175–1178.Google Scholar
Ka.Kadec, M. I.. Conditions on differentiability of the norm of a Banach space. Uspekhi Mat. Nauk., 20 (1965), 183–187.Google Scholar
K.Kwapien, S.. Isomorphic characterization of inner product space by orthogonal series with vector valued coefficients. Studia Math., 44 (1972), 583–595.CrossRefGoogle Scholar
KR.Kunen, K. and Rosenthal, H. P.. Martingale proofs of some geometrical results in Banach space theory. Pacific J. Math., 100 (1982), 153–177.Google Scholar
LW.Leach, E. B. and Whitfield, J. H. M.. Differentiable functions and rough norms on Banach spaces. Proc Amer. Math. Soc., 33 (1972), 120–126.CrossRefGoogle Scholar
LT.Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces, Vol. II. Function Spaces, (Springer-Verlag, 1979).Google Scholar
Me.Meshkov, V. Z.. Smoothness properties in Banach spaces. Studia Math., 63 (1978), 111–123.CrossRefGoogle Scholar
PWZ.Pechanec, J., Whitfield, J. H. M. and Zizler, V.. Norms locally dependent on finitely many coordinates. An. Acad. Brasil Gene, 53 (1981), 415–417.Google Scholar
P1.Phelps, R. R.. Convex functions, monotone operators and differentiability. Springer- Verlag Lecture Notes, 1264 (1989).CrossRefGoogle Scholar
P2.Phelps, R. R.. A representation theorem for bounded convex sets. Proc. Amer. Math. Soc., 11 (1960), 976–983.Google Scholar
R1.Rainwater, J.. Local uniform convexity of Day's norm on c0(T). Proc Amer. Math. Soc., 22 (1969), 335–339.Google Scholar
R2.Rainwater, J.. Day's norm on c0(T). Proc Functional Anal. Week, Various Publ. Series, 8 (Aarhus, 1969) 46–50.Google Scholar
S.Stegall, C.. Optimization of functions on certain subsets of Banach spaces. Math. Ann., 236 (1978), 171–176.CrossRefGoogle Scholar
TJ1.Tomczak-Jaegermann, N.. Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics 38 (Pitman, 1989).Google Scholar
TJ2.Tomczak-Jaegermann, N.. On the differentiability of the norm in the trace class Sp. Seminaire Maurey Schwartz (1974–1975).Google Scholar
Z.Zygmund, A.. Trigonometric Series. Vol. I (Cambridge University Press, 1959).Google Scholar