%0 Journal Article
%A Granero, A. S.
%A Jiménez Sevilla, María del Mar
%A Moreno, José Pedro
%T On the Nonseparable Subspaces of J(η) and C([1, η])
%D 2001
%@ 0025-584X
%U https://hdl.handle.net/20.500.14352/58624
%X 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 beingthe “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).
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