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)

 
 

 

On the von Neumann-Jordan constant
for Banach spaces


Authors: Mikio Kato and Yasuji Takahashi
Journal: Proc. Amer. Math. Soc. 125 (1997), 1055-1062
MSC (1991): Primary 46B20, 46B03, 46B42
DOI: https://doi.org/10.1090/S0002-9939-97-03740-4
MathSciNet review: 1371131
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $C_{\mathrm {NJ}} (E)$ be the von Neumann-Jordan constant for a Banach space $E$. It is known that $1\le C_{\mathrm {NJ}}(E)\le 2$ for any Banach space $E$; and $E$ is a Hilbert space if and only if $C_{\mathrm {NJ}} (E)=1$. We show that: (i) If $E$ is uniformly convex, $C_{\mathrm {NJ}} (E)$ is less than two; and conversely the condition $C_{\mathrm {NJ}} (E)<2$ implies that $E$ admits an equivalent uniformly convex norm. Hence, denoting by $\widetilde C_{\mathrm {NJ}} (E)$ the infimum of all von Neumann-Jordan constants for equivalent norms of $E$, $E$ is super-reflexive if and only if $\widetilde C_{\mathrm {NJ}} (E)<2$. (ii) If $\widetilde C_{\mathrm {NJ}} (E)=2^{2/p-1}$, $1<p\le 2$ (the same value as that of $L_p$-space), $E$ is of Rademacher type $r$ and cotype $r'$ for any $r$ with $1\le r<p$, where $1/r+1/r'=1$; the converse holds if $E$ is a Banach lattice and $l_p$ is finitely representable in $E$ or $E'$.


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

  • 1. B. Beauzamy, Introduction to Banach spaces and their geometry, 2nd Ed., North Holland, Amsterdam-New York-Oxford, 1985. MR 88f:46021
  • 2. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414.
  • 3. J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math. 38 (1937), 114-115.
  • 4. P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281-288. MR 49:1073
  • 5. E. Hewitt and K. Stromberg, Real and abstract analysis, Springer, New York-Heidelberg-Berlin, 1965. MR 32:5826
  • 6. R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542-550. MR 30:4139
  • 7. R. C. James, Super-reflexive Banach spaces, Canad. J. Math. 24 (1972), 896-904. MR 47:9248
  • 8. P. Jordan and J. von Neumann, On inner products in linear metric spaces, Ann. of Math. 36 (1935), 719-723.
  • 9. M. Kato and K. Miyazaki, Remark on generalized Clarkson's inequalities for extreme cases, Bull. Kyushu Inst. Tech., Math. Natur. Sci. 41 (1994), 27-31. MR 95h:46044
  • 10. M.Kato and K. Miyazaki, On generalized Clarkson's inequalities for $L_p(L_q)$ and Sobolev spaces, Math. Japon. 43 (1996), 505-515. CMP 96:13
  • 11. G. Köthe, Topologische lineare Räume I, Springer, Berlin-Heidelberg-New York, 1966. MR 33:3069
  • 12. J. Kuelbs, Probability on Banach spaces, Marcel Dekker, New York-Basel, 1978. MR 80c:60007
  • 13. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II, Springer, Berlin-Heidelberg-New York, 1979. MR 81c:46001
  • 14. B. Maurey and G. Pisier, Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach, Studia Math. 58 (1976), 45-90. MR 56:1388
  • 15. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350. MR 52:14940
  • 16. G. Pisier, Factorization of linear operators and geometry of Banach spaces, Amer. Math. Soc., Providence, RI, 1986. MR 88a:47020
  • 17. G. Pisier and Q. Xu, Random series in the real interpolation spaces between the spaces $v_p$, Lecture Notes in Math., vol. 1267, Springer-Verlag, Berlin-Heidelberg-New York, 1987, pp. 185-209. MR 89d:46011

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46B20, 46B03, 46B42

Retrieve articles in all journals with MSC (1991): 46B20, 46B03, 46B42


Additional Information

Mikio Kato
Affiliation: Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804, Japan

Yasuji Takahashi
Affiliation: Department of System Engineering, Okayama Prefectural University, Soja 719-11, Japan

DOI: https://doi.org/10.1090/S0002-9939-97-03740-4
Keywords: von Neumann-Jordan constant, uniform convexity, super-reflexivity, type and cotype, finite representability, $p$-convexity and $p$-concavity for a Banach lattice
Received by editor(s): September 8, 1995
Additional Notes: The authors were supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (07640225 (first author), 07640240 (second author))
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society