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Weakly uniformly rotund Banach spaces. (English) Zbl 1060.46502

A Banach space \(E\) is said to be weakly uniformly rotund (WUR), if for any sequences \((x_n)\) and \((y_n)\) in the unit sphere with \(\| x_n+y_n\| \to 2\) we have that weak-\(\lim _n(x_n-y_n)=0\). A topological space \(A\) is \(K\)-analytic, if it is a continuous image of a closed subset of a product \(K\times M\), where \(K\) is compact and \(M\) is a Polish space. Let \(E\) be a WUR Banach space. The authors show that then the dual space \(E^*\) is \(K\)-analytic in the weak topology. As a corollary they get that every WUR Banach space is an Asplund space (a Banach space on which every convex continuous function is Fréchet differentiable on a dense set); this was originally proved by a different method in P. Hájek [Comment. Math. Univ. Carolin. 37, 241-253 (1996; Zbl 0855.46005)]. As another corollary they get the result of M. Fabian, P. Hájek and V. Zizler [Serdica Math. J. 23, No. 3–4, 351–362 (1997; Zbl 0974.46025)] that every WUR Banach space admits an equivalent locally uniformly rotund norm.

MSC:

46B20 Geometry and structure of normed linear spaces
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