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Normal structure for Banach spaces with Schauder decomposition. (English) Zbl 0647.46016
The normal structure property has got a flourishing development in the last decade. The obvious reason is its connection to fixed point property. In this paper we associate to any Banach space X, with Schauder decomposition, a constant \(\beta _ p(X)\) for \(p\in [1,\infty)\). The main result states that X has weak-normal structure provided \(\beta _ p(X)<2^{1/p}\). We also discuss the relation between weak\({}^ *\)- normal structure and \(\beta _ p(X)\) in dual case. So we gave a positive answer to Kirk-Soardi’s problem by proving that a dual space X has \(weak^ *\)-normal structure provided \(d(X,\ell _ 1)<2\) (where d is the Banach-Mazur distance).
Reviewer: M.A.Khamsi

MSC:
46B20 Geometry and structure of normed linear spaces
47H10 Fixed-point theorems
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