Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-01T06:45:51.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform asymptotic smoothness of norms

Published online by Cambridge University Press:  17 April 2009

T. Lewis ,
J. Whitfield  and
V. Zizler
T. Lewis
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.
J. Whitfield
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada.
V. Zizler
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a notion of smoothness of a norm on a Banach space X which generalizes the notion of uniform differentiability and is formulated in terms of unicity of Hahn Banach extensions of functionals on block subspaces of a fixed Schauder basis S in X. Variants of this notion have already been used in estimating moduli of convexity in some spaces or in fixed point theory. We show that the notion can also be used in studying the convergence of expansions coefficient of elements of X* along the dual basis S*.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 3 , June 1986 , pp. 457 - 470
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Altshuler, Z., “Uniform convexity in Lorentz sequence spaces”, Israel J. Math. 20 (1975), 260–274.CrossRefGoogle Scholar
[2]Bui-Minh-Chi, and Guraii, V.I., “Some characteristics of normed spaces and their applications to the generalization of Parseval's inequality for Banach spaces”, Teor. Funkoiǐ Funkeional. Anal. i Priložen 8 (1969), 74–91 (Russian).Google Scholar
[3]Day, M. M., Normed linear spaces (Springer Verlag, 1975).Google Scholar
[4]Diestel, J., Geometry of Banach spaces – selected topics (Lecture Notes in Mathematics 485. Springer Verlag, 1975).CrossRefGoogle Scholar
[5]van Dulst, D., and Sims, B., Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), in Banach space theory and its applications, Proceedings, Bucharest 1981, (Lecture Notes in Mathematics 991. Springer Verlag, 1983).Google Scholar
[6]Figiel, T., “On the moduli of convexity and smoothness”, Studia Math. 56 (1976), 121–155.CrossRefGoogle Scholar
[7]Giles, J., Convex analysis with applications in differentiation of convex functions, (Res. Notes in Math. 58. Pitnam, 1982).Google Scholar
[8]Gossez, J.-P. and Dozo, E. Lami, “Structure normale et base de Schauder”, Acad. Roy. Belg. Bull. Cl. Sci. 55 (1969), 673–681.Google Scholar
[9]Huff, R., “Banach spaces which are nearly uniformly convex”. Rocky Mountain J. Math. 10 (1980), 743–749.CrossRefGoogle Scholar
[10]Istratescu, V., Strict convexity and complex strict convexity, theory and applications. (Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, 1984).Google Scholar
[11]Kadec, M. I., “Unconditional convergence of series in uniformly convex spaces”, Uspekhi Mat. Nauk (N.S.) 11 (1956), 185–190 (Russian).Google Scholar
[12]Kothe, G., Topological vector spaces I (English translation), (Springer Verlag, 1969).Google Scholar
[13]Linderstrauss, J. and Tzafriri, L., Classical Banach spaces I, sequence spaces (Springer Verlag, 1977).Google Scholar
[14]Linderstrauss, J. and Tzafrinzi, L., Classical Banach spaces II, function spaces (Springer Verlag, 1979).CrossRefGoogle Scholar
[15]Lovaglia, A. R., “Locally uniformly convex spaces”, Trans. Amer. Math. Soc. 78 (1955), 225–238.CrossRefGoogle Scholar
[16]Pisier, G., “Martingales with values in uniformly convex spaces”, Israel J. Math. 20 (1975), 326–350.CrossRefGoogle Scholar