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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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