Weak and quasi approximation properties in Banach spaces
type: Item
instance of: scholarly article
title: Weak and quasi approximation properties in Banach spaces
zbMATH Open document ID: 1098.46016
author: Changsun Choi
author: Ju Myung Kim
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: The authors say that a Banach space \(X\) has the [bounded] weak approximation property if every compact operator \(T: X\to X\) is the limit of a [bounded] net of finite-rank operators for the topology of uniform convergence on compact sets, and they say that \(X\) has the quasi approximation property if the convergence takes place with respect to the operator norm. The paper presents some observations concerning these new notions that are elementary, well known or follow the same proofs as their classical counterparts (e.g., the weak approximation property of \(X^*\) implies that of \(X\)). No result is nontrivially new, and what the authors call their own proof of Lemma~3.1, a result due to Grothendieck, is actually a copy of the proof of Proposition~1.e.3 in \textit{J.~Lindenstrauss}' and \textit{L.~Tzafriri}'s classical book [Classical Banach Spaces~I. Springer-Verlag (1977; Zbl 0362.46013); re-edited 1996 (Zbl 0852.46015)]. For more serious recent attempts to weaken the approximation property see [\textit{E.~Odell, H.-O.~Tylli}, Trans.\ Am.\ Math.\ Soc.\ 357, No. 3, 1125--1159 (2005; Zbl 1067.46028)] or [\textit{Å.~Lima, E.~Oja}, Math.\ Ann.\ 333, No. 3, 471--484 (2005; Zbl 1097.46012)].
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B28
zbMATH DE Number: 5016213
zbMATH Keywords: approximation property
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.05.013
OpenAlex ID: W1993917821
cites work: Approximation properties.
cites work: Produits tensoriels topologiques et espaces nucléaires
cites work: On the Existence of Strongly Series Summable Markuschevich Bases in Banach Spaces
cites work: Spaces of compact operators
cites work: Q4136031
cites work: An Introduction to Banach Space Theory
cites work: The Compact Approximation Property does not imply the Approximation Property
DOI: 10.1016/J.JMAA.2005.05.013
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description: scientific article; zbMATH DE number 5016213