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Valdivia compacta and equivalent norms. (English) Zbl 1073.46009

Summary: We prove that the dual unit ball of a Banach space \(X\) is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on \(X\) is a Valdivia compactum. As a corollary, we show that the dual unit ball of a Banach space \(X\) of density \(\aleph_1\) is Corson if (and only if) \(X\) has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler [Bull. Pol. Acad. Sci., Math. 47, 221–230 (1999; Zbl 0946.46016)] and yield a converse to D. Amir’s and J. Lindenstrauss’s theorem [Ann. Math. (2) 88, 35–46 (1968; Zbl 0164.14903)].

MSC:

46B26 Nonseparable Banach spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
54D30 Compactness
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