The Lindelöf property in Banach spaces

B. Cascales ; I. Namioka ; J. Orihuela

Studia Mathematica, Tome 157 (2003), p. 165-192 / Harvested from The Polish Digital Mathematics Library

Résumé

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$, where, for each S ⊂ D, $d_{S}(x,y) = sup{ϱ(x(t),y(t)): t∈ S}$. (ii) For each countable subset A of D, $(K,d_{A})$ is separable.i (iii) The space (K,γ(D)) is Lindelöf, where γ(D) is the topology of uniform convergence on the family of countable subsets of D. (iv) $(K,γ(D))^{{ℕ}}$ is Lindelöf. The rest of the paper is devoted to applications of the basic theorem. Here are some of them. A compact Hausdorff space K is Radon-Nikodým compact if, and only if, there is a bounded subset D of C(K) separating the points of K such that (K,γ(D)) is Lindelöf. If X is a Banach space and H is a weak*-compact subset of the dual X* which is weakly Lindelöf, then $(H,{weak})^{ℕ}$ is Lindelöf. Furthermore, under the same condition $\overline{span(H)}^{|| ||}$ and $\overline{co(H)}^{w*}$ are weakly Lindelöf. The last conclusion answers a question by Talagrand. Finally we apply our basic theorem to certain classes of Banach spaces including weakly compactly generated ones and the duals of Asplund spaces.

Détail

Publié le : 2003-01-01

Zbl 1038.54008

EUDML-ID : urn:eudml:doc:285329