Complemented Copies of l1 and Pełczyński's Property (V *) in Bochner Function Spaces

Randrianantoanina, Narcisse

Canadian journal of mathematics, Tome 48 (1996) no. 3, pp. 625-640

Résumé

Let X be a Banach space and (f n )n be a bounded sequence in L 1(X). We prove a complemented version of the celebrated Talagrand's dichotomy, i.e., we show that if (en )n denotes the unit vector basis of c 0, there exists a sequence gn ∈ conv(f n , f n+1,...) such that for almost every ω, either the sequence (gn (ω) ⊗ en ) is weakly Cauchy in or it is equivalent to the unit vector basis of l 1. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of l 1 in L 1(X). As an application, we show that for a Banach space X, the space L 1(X) has Pełczyńiski's property (V*) if and only if X does.

Détail

DOI : 10.4153/CJM-1996-033-2

Mots-clés : 46E40, 28B05, 28B20, Weakly compact sets, Bochner function spaces

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