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2003 | 154 | 2 | 165-192
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The Lindelöf property in Banach spaces

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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.
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  • Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo Murcia, Spain
autor
  • Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195-4350, U.S.A.
autor
  • Departamento de Matemáticas, Universidad de Murcia, 30.100 Espinardo Murcia, Spain
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-2-4
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