The Lindelöf property in Banach spaces

B. Cascales¹ ; I. Namioka² ; J. Orihuela³

¹Departamento de Matemáticas Universidad de Murcia 30100 Espinardo Murcia, Spain
²Department of Mathematics, Box 354350 University of Washington Seattle, WA 98195-4350, U.S.A.
³Departamento de Matemáticas Universidad de Murcia 30.100 Espinardo Murcia, Spain

Studia Mathematica, Tome 154 (2003) no. 2, pp. 165-192

Résumé

A topological space $(T,\tau)$ 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 $(M,\varrho)$ be a metric space with $\varrho$ 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 $S\subset D$, $$d_{S}(x,y)=\sup\{\varrho(x(t),y(t)): t\in S\}.$$(ii) For each countable subset $A$ of $D$, $(K,d_{A})$ is separable.(iii) The space $(K,\gamma (D))$ is Lindelöf, where $\gamma (D)$ is the topology of uniform convergence on the family of countable subsets of $D$.(iv) $(K,\gamma (D))^{{\mathbb N}}$ 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 $(K,\gamma (D))$ is Lindelöf. If $X$ is a Banach space and $H$ is a weak$^{\ast}$-compact subset of the dual $X^\ast$ which is weakly Lindelöf, then $(H,\hbox{weak})^{\mathbb N}$ is Lindelöf. Furthermore, under the same condition $\overline{{\rm span}(H)}^{\,\|\ \|}$ and $\overline{{\rm co}{(H)}}^{\,w^{\ast}}$ 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

DOI : 10.4064/sm154-2-4

Keywords: topological space tau said fragmented metric each non empty subset has non empty relatively subsets arbitrarily small d diameter basic theorem present paper following varrho metric space varrho bounded arbitrary index set compact subset product space following conditions equivalent fragmented where each subset sup varrho each countable subset separable iii space gamma lindel where gamma topology uniform convergence family countable subsets gamma mathbb lindel rest paper devoted applications basic theorem here compact hausdorff space radon nikod compact only there bounded subset separating points gamma lindel banach space weak ast compact subset dual ast which weakly lindel hbox weak mathbb lindel furthermore under condition overline span overline ast weakly lindel conclusion answers question talagrand finally apply basic theorem certain classes banach spaces including weakly compactly generated duals asplund spaces