Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Skewness in Banach spaces


Authors: Simon Fitzpatrick and Bruce Reznick
Journal: Trans. Amer. Math. Soc. 275 (1983), 587-597
MSC: Primary 46B20; Secondary 46C05
DOI: https://doi.org/10.1090/S0002-9947-1983-0682719-5
MathSciNet review: 682719
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $E$ be a Banach space. One often wants to measure how far $E$ is from being a Hilbert space. In this paper we define the skewness $s(E)$ of a Banach space $E$, $0 \leqslant s(E) \leqslant 2$, which describes the asymmetry of the norm. We show that $s(E) = s({E^{\ast }})$ for all Banach spaces $E$. Further, $s(E) = 0$ if and only if $E$ is a (real) Hilbert space and $s(E) = 2$ if and only if $E$ is quadrate, so $s(E) < 2$ implies $E$ is reflexive. We discuss the computation of $s({L^p})$ and describe its asymptotic behavior near $p = 1,2$ and $\infty$. Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.


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

  • Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
  • Mahlon M. Day, Normed linear spaces, 3rd ed., Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. MR 0344849
  • Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 173932, DOI https://doi.org/10.2307/1970663
  • R. K. Ritt, A generalization of inner product, Michigan Math. J. 3 (1955), 23–26. MR 70972

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46B20, 46C05

Retrieve articles in all journals with MSC: 46B20, 46C05


Additional Information

Article copyright: © Copyright 1983 American Mathematical Society