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Banach space properties sufficient for normal structure. (English) Zbl 1133.46009

In [Pac.J.Math.86, 427–436 (1980; Zbl 0442.46018)], W.L.Bynum introduced a coefficient \(WCS(X)\) based on weakly convergent sequences in a Banach space \(X\) that could be used to imply that a Banach space has weak normal structure. In the article under review, the author establishes some lower bounds for \(WCS(X)\) and uses these lower bounds to improve the results of several authors. For example, in [Nonlinear analysis and applications, Proc.int.Conf., St.John’s/Newfoundland 1981, Lect.Notes Pure Appl.Math.80, 279–286 (1982; Zbl 0499.47034)], the reviewer showed that, if the modulus of smoothness of a Banach space satisfies \(\rho_X'(0)< 1/2\), then \(X\) has normal structure. The author improves this result by showing that \(X\) has normal structure if \(\rho_X'(0)< M(X)/2\), where \(M(X)\) is a coefficient introduced by T.Domínguez–Benavides [Houston J. Math.22, No.4, 835–849 (1996; Zbl 0873.46012)]. The author also provides stronger sufficient conditions than previously known for a Banach space to have normal structure using the James constant \(J(X)\), the von Neumann–Jordan constant \(C_{NJ}(X)\), and several other constants.

MSC:

46B20 Geometry and structure of normed linear spaces
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