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

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 c 0 (η) complemented in C ([1, η]). In particular, every nonseparable closed subspace of J (ω 1 ) (resp. C ([1, ω 1 ])) contains a complemented copy of 2 (ω 1 ) (resp. c 0 (ω 1 )). As consequence, we give examples ( J (ω 1 ), C ([1, ω 1 ]), 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).