Complemented Copies of ℓ1 and Pełczyński's Property (V*) in Bochner Function Spaces
1996 ◽
Vol 48
(3)
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pp. 625-640
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Keyword(s):
AbstractLet X be a Banach space and (fn)n be a bounded sequence in L1(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 c0, there exists a sequence gn ∈ conv(fn, fn+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 ℓ1. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of ℓ1 in L1(X). As an application, we show that for a Banach space X, the space L1(X) has Pełczyńiski's property (V*) if and only if X does.
1975 ◽
Vol 18
(1)
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pp. 137-140
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Keyword(s):
1975 ◽
Vol 20
(3-4)
◽
pp. 216-227
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Keyword(s):
Keyword(s):
1986 ◽
Vol 29
(3)
◽
pp. 329-333
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2007 ◽
Vol 75
(2)
◽
pp. 193-210
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Keyword(s):
1981 ◽
Vol 90
(1-2)
◽
pp. 63-70
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Keyword(s):