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On l 1 subspaces of Orlicz vector-valued function spaces.

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Bombal Gordón, Fernando (1987) On l 1 subspaces of Orlicz vector-valued function spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 101 (1). pp. 107-112. ISSN 0305-0041

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Official URL: http://journals.cambridge.org/abstract_S0305004100066445


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Abstract

The author studies those Orlicz vector-valued function spaces that contain a copy or a complemented copy of l 1 . Precisely, given a finite complete measure space (S,Σ,μ) , a Young function Φ , and a Banach space E , let L Φ (S,Σ,μ,E) denote the vector space of all (classes of) strongly measurable functions f from S to E such that ∫Φ(k∥f∥)dμ<∞ for some k>0 , and let L Φ (μ)=L Φ (S,Σ,μ,R) . The author first extends a result of G. Pisier concerning vector-valued L p function spaces by showing that if l 1 embeds in L Φ (S,Σ,μ,E) , then l 1 embeds either in L Φ (μ) or in E . This result, combined with a result of E. Saab and the reviewer concerning the embedding of l 1 as a complemented subspace of the Banach space of all E -valued continuous functions on a compact Hausdorff space, is used to show that if in addition E is a Banach lattice, if Φ satisfies the Δ 2 -condition and if μ is nonpurely atomic, then L Φ (S,Ω,μ,E) contains a complemented copy of l 1 if and only if either L Φ (μ) or E contains a complemented copy of l 1


Item Type:Article
Uncontrolled Keywords:Orlicz vector-valued;
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:18130
Deposited On:01 Feb 2013 12:10
Last Modified:02 Aug 2018 07:37

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