Atlas: The Lindelof property in Banach spaces by B. Cascales

Atlas home || Conferences | Abstracts | about Atlas

SumTopo 2001, Sixteenth Summer Conference on Topology and its Applications
July 18-21, 2001
City College of CUNY
New York, NY, USA

Organizers
Ralph Kopperman (City College, CUNY), Susan Andima (CW Post College, LIU), Gerald Itzkowitz (Queens College, CUNY), Prabudh Misra (College of Staten Island, CUNY), Shelly Rothman (CW Post College, LIU), Aaron Todd (Baruch College, CUNY)

View Abstracts
Conference Homepage

The Lindelöf property in Banach spaces
by
B. Cascales
Universidad de Murcia
Coauthors: I. Namioka (University of Washington), J. Orihuela (Universidad de Murcia)

A compact space K is fragmented by a metric d , when for every non-empty subset F of K and every \epsilon > 0 there is an open subset U in K such that U \cap F is non-empty and d-diameter-(U \cap F) is smaller than \epsilon. Given a compact subspace K of the cube [-1, 1]^D, we prove that K is fragmented with respect to the metric of uniform convergence on D if, and only if, K is Lindelof when endowed with the topology of uniform convergence on countable subsets of D. As consequences:

We prove that a dual Banach space X^* has the Radon-Nikodym property (RNP) iff X^* is Lindelöf with the topology of uniform convergence on bounded sequences of X, see [4]; we offer a topological version of this analytic result.

We give an alternative proof of Meyer's characterization of compact scattered spaces, see [3], by the Lindelöf property with respect to the G(delta)-topology.

We prove that if a dual Banach space X^* is weakly Lindelöf then, X^*×X^* is weakly Lindelöf -first proved in [4]-; we strengthen this result in several ways.

With the help of results in [1] too, we solve positively Problème 4.5 posed by Talagrand in [5] by proving: if X is a Banach space and H is a weak^*-compact subset of X^* which is weak-Lindelöf, then [`(co(H))]^{w^*} = [`(co (H))]^norm and this convex hull is weakly Lindelöf again;

We study Banach spaces X generated by compact sets H for vector topologies coarser than the weak topology, providing existence of PRI when H is fragmented by the norm. This gives a unified approach to the existence of PRI in WCG Banach spaces (Amir-Lindenstrauss) and in dual Banach spaces with the RNP;

We solve a problem by Mercourakis and Negrepontis posed in [2].

REFERENCES

B. Cascales, I. Namioka, and G. Vera. The Lindelöf property and fragmentability. Proc. Amer. Math. Soc., 128(11):3301-3309, 2000.

S. Mercourakis and S. Negrepontis. Banach spaces and topology. II. In Recent progress in general topology (Prague, 1991), pages 493-536. North-Holland, Amsterdam, 1992.

P. R. Meyer. Function spaces and the Aleksandrov-Uryshon conjecture. An. Mat. Pura and Appl., 86:25-29, 1970.

J. Orihuela. On weakly Lindelof Banach spaces. Progress in Functional Analysis Math.Studies North Holland, 170:279-291, 1992.

M. Talagrand, Deux generalisations d'un théorème de I.Namioka.Pacific J. Math., 81(1):239-251, 1979.

Date received: May 15, 2001