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On the Nonseparable Subspaces of J(η) and C([1, η])

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Granero, A. S. and Jiménez Sevilla, María del Mar and Moreno, José Pedro (2001) On the Nonseparable Subspaces of J(η) and C([1, η]). Mathematische Nachrichten, 221 (1). pp. 75-85. ISSN 0025-584X

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Official URL: http://onlinelibrary.wiley.com/doi/10.1002/1522-2616(200101)221:1%3C75::AID-MANA75%3E3.0.CO;2-T/abstract




Abstract

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 2(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c0(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω1) (resp. C([1,ω1])) contains a complemented copy of 2(ω1) (resp. c0(ω1)). As consequence, we give examples (J(ω1), C([1,ω1]), C(V ), V being
the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i. e., for every subspace Y ⊆ X we have that Dens (Y) = w∗ –Dens (Y ∗)), in spite of these spaces are not weakly Lindelof determined (WLD).


Item Type:Article
Uncontrolled Keywords:Long James spaces, Nonseparable subspaces.
Subjects:Sciences > Mathematics > Algebra
ID Code:22158
Deposited On:01 Jul 2013 17:13
Last Modified:21 Jun 2018 11:01

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