Projective resolutions, Markushevich bases, and equivalent norms
type: Item
instance of: scholarly article
title: Projective resolutions, Markushevich bases, and equivalent norms
zbMATH Open document ID: 0545.46010
DOI: 10.1007/BF01157451
published in: Mathematical Notes
publication date: 1983-01-01T00:00:00Z
review text: Let \(\omega\) (\(\Omega)\) be the first infinite (uncountable) ordinal and let dens X be the smallest cardinal c for which the Banach space X has a dense subset of cardinality c. Given an ordinal \(\alpha\), denote by \({\bar\alpha}\) its cardinal. The main result of this paper is the following theorem: Let \(P_{\beta}\), \(Q_{\beta}\), \(\omega \leq \beta \leq \Omega\), be two projective resolutions of the Banach space X (dens X\(=\Omega)\) such that \((*)\quad X^*=\cup_{\beta<\Omega}P^*_{\beta}X^*\). Then for each \(\beta_ 0<\Omega\) there exists an ordinal \(\gamma\) such that \(P_{\gamma}=Q_{\gamma}\), \(\beta_ 0<\gamma<\Omega.\) It is clear that if property (*) is fulfilled for one projective resolution then it is fulfilled for all projective resolutions. But, the author proves (Th. 2) that the property (*) is not allways fulfilled even for Banach spaces with M-bases; for instance the space \(C_ 0(\Omega)\). From the main theorem, certain existence results of M-bases are derived. Thus, one of this results, which at the same time solves a problem from \textit{I. Singer's} book ''Bases in Banach Spaces. II.'' (1981; Zbl 0467.46020), (Problem 20.8, p. 837) says: If a Banach space has a locally uniformly convex norm and a weak* angelic dual, then projection resolutions and M-bases do not exist.
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B10
zbMATH DE Number: 3867992
zbMATH Keywords: Markushevich bases
zbMATH Keywords: equivalent norms
zbMATH Keywords: projective resolutions
zbMATH Keywords: Banach spaces with M-bases
zbMATH Keywords: locally uniformly convex norm
zbMATH Keywords: \(weak^*\) angelic dual
author: A. M. Plichko
MaRDI profile type: Publication
cites work: Q3920145
cites work: Markusevic Bases in Some Dual Spaces
cites work: Q3851121
cites work: Sur un théorème de Banach
cites work: Examples of separable spaces which do not contain $l_{1}$ and whose duals are non-separable
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cites work: Q4134417
description: scientific article; zbMATH DE number 3867992