0 O ct 1 99 2 WEAKLY LINDELOF DETERMINED BANACH SPACES NOT CONTAINING
نویسنده
چکیده
The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing l(N). Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing l(N) and not being itself a W.C.G. space. INTRODUCTION In the present paper we deal with Banach spaces not containing isomorphically the space l(N). The motivation for this study was a problem, posed to us by S. Merkourakis, related to the heredity problem for weakly compactly generated (W.C.G.) Banach spaces. The problem in question is the following: Is every W.C.G. Banach space X not containing l(N) hereditarily W.C.G.? It is well known by a classical example due to Rosenthal [R] that the heredity problem for W.C.G. has negative answer. On the other hand, M. Fabian has shown in [F] that if the conjugate X satisfies the Radon-Nikodym property (R.N.P.) then the space X is hereditarily W.C.G. Actually M. Fabian proved the stronger result that if X is weakly countably determined (W.C.D.) and X satisfies RNP then X is W.C.G. space. Later M. Valdivia in [V] extended this result to the class of weakly
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