Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-01T18:31:50.497Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Parallelogram Laws for Banach Spaces

Published online by Cambridge University Press:  20 November 2018

W. L. Bynum
W. L. Bynum*
Affiliation:
College of William and Mary Williamsburg, Virginia 23185, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

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.

It has been shown previously that the Lp(μ) spaces for 1 < p ≤ 2 satisfy a weak parallelogram law, and the same methods can be used to show that the Lp(μ) spaces for 2 ≤ p <∞ satisfy a related weak parallelogram law. This paper obtains several equivalent characterizations of Banach spaces which satisfy one of these two weak parallelogram laws. One such characterization involves the conditions on the moduli of convexity and smoothness analyzed by Lindenstrauss.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 3 , 01 September 1976 , pp. 269 - 275
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Bynum, W. L. and Drew, J. H., A weak parallelogram law for ℓp , Amer. Math. Monthly 79 (1972), 1012–1015.Google Scholar
2. Day, M. M., Some characterizations of inner product spaces, Trans. Amer. Math. Soc. 62 (1947), 320–337.Google Scholar
3. DePrima, C. R. and Petryshyn, W. V., Remarks on strict monotonicity and surjectivity properties of duality mappings defined on real normed linear spaces, Math. Z. 123 (1971), 49–55.Google Scholar
4. Figiel, T. and Pisier, G., Séries aléatoires dans les espaces uniformément convexes ouuniformḿent lisses, C. R. Acad. Sci. Paris Ser. A 279 (1974), 611–614.Google Scholar
5. Hanner, O., On the uniform convexity of Lp and ℓp , Ark. Mat. 3 (1956), 239–244.Google Scholar
6. Jordan, P. and J. von Neumann, On inner products in linear metric spaces, Ann. of Math. (2) 36 (1935), 719–723.Google Scholar
7. Lindenstrauss, J., On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J. 10 (1963), 241–252.Google Scholar
8. Schoenberg, I. J., A remark on M. M. Day’s characterization of inner product spaces and aconjecture of L. M. Blumenthal, Proc. Amer. Math. Soc. 3 (1952), 961–964.Google Scholar