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The characteristic of convexity of a Banach space and normal structure. (English) Zbl 1129.46010

The author presents some sufficient conditions for a Banach space or its dual to have normal (or uniform normal) structure in terms of several well-known parameters. For example, the author proves that, if \(J(X^*)\, \varepsilon_0(X) < 2\), then \(X^*\) has uniform normal structure. (Here \(\varepsilon_0(X)\) denotes the characteristic of convexity of \(X\), roughly speaking, the length of the longest line segment that can be put arbitrarily close to the unit sphere of \(X\); and \(J(X)\) denotes the James constant of \(X\), a parameter used in determining if a Banach space is uniformly nonsquare.) As a corollary, the author notes that, if \(\varepsilon_0(X) = 1\), then \(X^*\) has uniform normal structure. Prior to this, it was only known that, if \(\varepsilon_0(X) < 1\), then \(X^*\) has uniform normal structure [S.Prus, Bull.Pol.Acad.Sci., Math.36, No.5–6, 225–227 (1988; Zbl 0767.46009)]. The constant \(1\) in the author’s result is the best possible. The author also gives conditions in terms of the characteristic of convexity and a coefficient of weak orthogonality for a Banach space or its dual to have normal structure.

MSC:

46B20 Geometry and structure of normed linear spaces

Citations:

Zbl 0767.46009
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References:

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