On the nonseparable subspaces of J(eta) and C([1,eta])

Let eta be a regular cardinal. It is proved, among other things, that: (i) if J(eta) is the corresponding long James space, then every closed subspace Y subset of or equal to J(eta), with Dens(Y) = eta, has a copy of l(2)(eta ) complemented in J(eta); (ii) if Y is a closed subspace of the space of co ntinuous functions C([1, eta]), with Dens(Y) = eta, then Y has a copy of c( 0)(eta) complemented in C([1,eta]). In particular, every nonseparable close d subspace of J(omega (1)) (resp. C([1,omega (1)])) contains a complemented copy of l(2)(omega (1)) (resp, c(0)(omega (1))). As consequence, we give e xamples (J(omega (1)), C([1,omega (1)]), C(V), V being the "long segment") of Banach spaces X with the hereditary density property (HDP) (i. e., for e very subspace Y subset of or equal to X are have that Dens(Y) = w* -Dens(Y* )), in spite of these spaces are not weakly Lindelof determined (WLD).