Fragmentability in Banach spaces: interaction of topologies
type: Item
instance of: scholarly article
title: Fragmentability in Banach spaces: interaction of topologies
zbMATH Open document ID: 1237.54002
DOI: 10.5052/RACSAM.2010.18
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2011-07-18T00:00:00Z
review text: This paper is a survey article aimed to give a relatively concise description of the state-of-the-art of the theory of fragmentability and \(\sigma\)-fragmentability in Banach spaces for a broad audience. Modern directions of investigation, related to renorming of Banach spaces and the Lindelöf property are also discussed. Here, the author concentrates mainly on classical problems of the theory, including different characterizations for a Banach space to have an equivalent Kadec (resp. LUC) norm, the existence of a \(\tau_p\)-Kadec norm on \(C(K)\) when \(K\) is a compact space and the characterization of fragmentability through the Lindelöf property. Most proofs are given and most of them are accompanied with historical references and comments about open questions for future research. A comprehensive bibliography is provided at the end of the paper. The first section, of preliminary nature, is dedicated to presenting the basic results and definitions for the rest of the paper; among other things, the Radon-Nikodým Property (RNP), Asplund spaces and an application to the Ryll-Nardzewski's fixed point theorem are presented. Section 2 is one of the main parts of the paper. Here, it is proved that the fragmentability of a compact subset \(K\) of \([-1,1]^D\) is equivalent to the separability of the metric spaces \((K,d_A)\) for each countable set \(A\subseteq D\). As a consequence, it follows easily that each \(\tau_p\)-compact subset of \(C(K)\) is norm-fragmented for every compact Hausdorff space \(K\) and that each \(w\)-compact subset of a Banach space is norm-fragmented. Radon-Nikodým (RN) compacta are defined as those compact spaces that are homeomorphic to a weak\(^\ast\)-compact subset of a dual Banach space with the RNP. As a consequence, one obtains that Eberlein compact and scattered compact Hausdorff spaces are RN-compact, and that a compact space \(K\) is RN-compact iff \(K\) is fragmented by a lower-semicontinuous metric. From this fact, it follows that \…
reviewed by: Salvador Hernández
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54H99
zbMATH DE Number: 5925092
zbMATH Keywords: fragmentability
zbMATH Keywords: \(\sigma\)-fragmentability
zbMATH Keywords: renorming
author: Isaac Namioka
MaRDI profile type: Publication
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full work available at URL: racsam.2010.18
OpenAlex ID: W1584094073
description: scientific article; zbMATH DE number 5925092