TY - JOUR

T1 - Skewness in banach spaces

AU - Fitzpatrick, Simon

AU - Reznick, Bruce

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1983/2

Y1 - 1983/2

N2 - 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 ≤ s(E) ≤ 2, which describes the asymmetry of the norm. We show that s(E) = s(E*) for all Banach spaces E. Further, s(E) = 0 if and only if E is a (real) (Formula presnt) Hilbert space and s(E) = 2 if and only if E is quadrate, so 5(E) < 2 implies £ is reflexive. We discuss the computation of s(LP) and describe its asymptotic behavior near p = 1, 2 and ∞ Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.

AB - 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 ≤ s(E) ≤ 2, which describes the asymmetry of the norm. We show that s(E) = s(E*) for all Banach spaces E. Further, s(E) = 0 if and only if E is a (real) (Formula presnt) Hilbert space and s(E) = 2 if and only if E is quadrate, so 5(E) < 2 implies £ is reflexive. We discuss the computation of s(LP) and describe its asymptotic behavior near p = 1, 2 and ∞ Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.

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U2 - 10.1090/S0002-9947-1983-0682719-5

DO - 10.1090/S0002-9947-1983-0682719-5

M3 - Article

AN - SCOPUS:84967773287

VL - 275

SP - 587

EP - 597

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -