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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
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.

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

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Article copyright: © Copyright 1983 American Mathematical Society