On the Nonseparable Subspaces of . . .
نویسنده
چکیده
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 c 0 () 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. c 0 (! 1)). As a 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)), even though these spaces are not weakly countably determined (WCD). 1. Notations and preliminaries Throughout, (X; k k) will be a real Banach space, B(X) the closed unit ball of X, S(X) the unit sphere and X its topological dual. Let O denote the ordinal numbers, LO the limit ordinals and C the cardinals. If 2 C, the coonality cf() of is the smallest cardinal for which there exists a sequence of ordinals f i g 1i<< ; i 2 O; i < and = supf i : 1 i < g. A cardinal is said to be regular if cf() =. Denote by RC the family of regular cardinals. If A; B are subsets of the ordinal , we write A < B ii, 8 2 A; 8 2 B, we have <. A transsnite sequence fA g 1<< ; 2 O, of subsets of is said to be a skipped (transsnite) sequence ii A < A , whenever < < , and for each < ; 9n < such that A < n < A +1. A subset A is said to be nice if min(A) = 2 LO. We say that S is a segment of if S and ; ] S whenever ; 2 S; .
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