The Lindelöf property in Banach spaces
Volume 154 / 2003
Studia Mathematica 154 (2003), 165-192
MSC: Primary 46B99, 46A50; Secondary 54C35.
DOI: 10.4064/sm154-2-4
Abstract
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.
