The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, and J. Orihuela. "The Lindelöf property in Banach spaces." Studia Mathematica 154.2 (2003): 165-192. <http://eudml.org/doc/285329>.

@article{B2003,
abstract = {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.},
author = {B. Cascales, I. Namioka, J. Orihuela},
journal = {Studia Mathematica},
keywords = {Lindelöf space; fragmentability; Radon-Nikodým compact space; -topology},
language = {eng},
number = {2},
pages = {165-192},
title = {The Lindelöf property in Banach spaces},
url = {http://eudml.org/doc/285329},
volume = {154},
year = {2003},
}

TY - JOUR
AU - B. Cascales
AU - I. Namioka
AU - J. Orihuela
TI - The Lindelöf property in Banach spaces
JO - Studia Mathematica
PY - 2003
VL - 154
IS - 2
SP - 165
EP - 192
AB - 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.
LA - eng
KW - Lindelöf space; fragmentability; Radon-Nikodým compact space; -topology
UR - http://eudml.org/doc/285329
ER -