Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Interpolation of uniformly convex Banach spaces


Authors: Michael Cwikel and Shlomo Reisner
Journal: Proc. Amer. Math. Soc. 84 (1982), 555-559
MSC: Primary 46M35; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1982-0643748-5
MathSciNet review: 643748
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $ {A_0}$ and $ {A_1}$ are a compatible couple of Banach spaces, one of which is uniformly convex, then the complex interpolation spaces $ {[{A_0},{A_1}]_\theta }$ are also uniformly convex for $ 0 < \theta < 1$. Estimates are given for the moduli of convexity and smoothness of $ {[{A_0},{A_1}]_\theta }$ in terms of these moduli for $ {A_0}$ and $ {A_1}$. In general, up to equivalence of moduli these estimates are best possible.


References [Enhancements On Off] (What's this?)

  • [1] B. Beauzamy, Espaces d'interpolation réels: Topologie et géométrie, Lecture Notes in Math., vol. 666, Springer-Verlag, Berlin and New York, 1978. MR 513228 (80k:46080)
  • [2] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Math. Wissenschaften, Bd. 223, Springer-Verlag, Berlin and New York, 1976. MR 0482275 (58:2349)
  • [3] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190. MR 0167830 (29:5097)
  • [4] T. Figiel, On the moduli of convexity and smoothness, Studia Math. 56 (1976), 121-155. MR 0425581 (54:13535)
  • [5] T. Figiel and G. Pisier, Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses, C. R. Acad. Sci. Paris Ser. A 279 (1974), 611-614. MR 0358295 (50:10761)
  • [6] D. J. Garling, Convexity, smoothness and martingale inequalities, Israel J. Math. 29 (1978), 189-198. MR 0473788 (57:13450)
  • [7] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Parts I, II, Ergebnisse der Math. und ihrer Grenzgebiete, Bds. 92, 97, Springer-Verlag, Berlin and New York, 1977, 1979. MR 0415253 (54:3344)
  • [8] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350. MR 0394135 (52:14940)
  • [9] -, Some applications of the complex interpolation method to Banach lattices, J. Analyse Math. 35 (1979), 264-280. MR 555306 (80m:46020)
  • [10] -, La méthode d'interpolation complexe: Application aux treillis de Banach, École Polytechnique Séminaire d'Analyse Fonctionelle, 1978-79, Exp. 17.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46M35, 46B20

Retrieve articles in all journals with MSC: 46M35, 46B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0643748-5
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society