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Complemented Copies of ℓ1 and Pełczyński's Property (V *) in Bochner Function Spaces
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- Journal:
- Canadian Journal of Mathematics / Volume 48 / Issue 3 / 01 June 1996
- Published online by Cambridge University Press:
- 20 November 2018, pp. 625-640
- Print publication:
- 01 June 1996
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- Article
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- You have access
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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.
or it is equivalent to the unit vector basis of 