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Resumen:
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Let ? be a regular cardinal. It is proved, among other things, that: (i) if J(?) is the corresponding long James space, then every closed subspace Y ? J(?), with Dens (Y) = ?, has a copy of 2(?) complemented in J(?); (ii) if Y is a closed subspace of the space of continuous functions C([1, ?]), with Dens (Y) = ?, then Y has a copy of c0(?) complemented in C([1, ?]). In particular, every nonseparable closed subspace of J(?1) (resp. C([1,?1])) contains a complemented copy of 2(?1) (resp. c0(?1)). As consequence, we give examples (J(?1), C([1,?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).
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![On the Nonseparable Subspaces of J(?) and C([1, ?])](./styles/zen/images/no_image.jpg)
