Abstract: The purpose of this paper is to show how the method, developed by M. Valdivia, of constructing projections in a weakly compactly generated Banach space X can be used to give a direct proof of the Lindelöf property for the weak topology in X. Since this method extends to large classes of spaces we prove a Lindelöf property for Banach spaces with a Valdivia compact weak* dual unit ball. Our result includes previous ones of K. Alster, R. Pol and S. P. Gul'ko for the space C(K) with the topology of pointwise convergence, where K is a Corson compact space. If X is a Banach space then the dual X* has the Radon-Nikodym property if and only if it is Lindelöf in the topology of uniform convergence on the separable bounded subsets of X. Finally, if a dual space X* is weakly Lindelöf then the bidual unit ball must be a Corson compact space in the weak* topology, so the product X* ×X* is weakly Lindelöf, too. This result gives a positive answer to a problem of Corson in the case of dual spaces.
Publication Year: 1992
Publication Date: 1992-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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Cited By Count: 21
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