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

Published online by Cambridge University Press:  20 November 2018

Narcisse Randrianantoanina
Narcisse Randrianantoanina*
Affiliation:
Department of Mathematics The University of Texas at Austin Austin, Texas 78712 U.S.A. email: nrandri@math.utexas.edu
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Abstract

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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.

Keywords

46E4028B0528B20Weakly compact setsBochner function spaces

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 3 , 01 June 1996 , pp. 625 - 640
Copyright
Copyright © Canadian Mathematical Society 1996

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