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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
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0682719-5
Article copyright: © Copyright 1983 American Mathematical Society