Hilbert generated Banach spaces need not have a norming Markushevich basis
type: Item
instance of: scholarly article
title: Hilbert generated Banach spaces need not have a norming Markushevich basis
zbMATH Open document ID: 1445.46017
author: Petr Hájek
published in: Advances in Mathematics
publication date: 2019-07-10T00:00:00Z
review text: Weakly compactly generated (for short, WCG) Banach spaces (i.e., spaces with a weakly compact linearly dense subset, e.g., separable spaces, reflexive spaces, \(c_0(\Gamma)\) for every \(\Gamma\), \(L_1(\mu)\) for a \(\sigma\)-finite measure \(\mu\), \(\ldots\)) form a fundamental class in the study of general (nonseparable) Banach spaces. The presence of a kind of ``coordinate system'' (Schauder bases and, more generally, Markushevich (M, for short) bases) throws a strong light on the structure of the space and allows, in particular, for renormings. WCG Asplund spaces are characterized as Banach spaces admitting a shrinking M-basis (i.e., the closed linear span \(F\) of the set of all functional coefficients is the whole dual). If the supremum on the closed unit ball of \(F\) gives an equivalent norm on the space, the M-basis is said to be norming (\(1\)-norming if it gives the original norm). It has been an important open problem for fifty years whether every WCG Banach space admits a norming M-basis. The present paper solves this problem in the negative. Amazingly, the counterexample enjoys a bunch of extra properties: it is a \(C(K)\)-space, where \(K\) ia a zero-dimensional uniform Eberlein compact, so the space is Hilbert generated -- i.e., there exists a bounded linear operator from some \(\ell_2(\Gamma)\) into \(C(K)\) with dense range. The last part of the paper shows that the space \(C(\sigma_1(\Gamma)^{\omega})\) has a \(1\)-norming M-basis (here, \(\sigma_1(\Gamma)\) is the one-point compactification of a discrete set \(\Gamma\), and \(\sigma_1(\Gamma)^{\omega}\) is just the Cartesian product of \(\omega\) copies of this space). This provides a test for checking whether a compact space \(K\) is a continuous image of \(\sigma_1(\Gamma)^{\omega}\). Another consequence is that having a norming M-basis for a Banach space \(C(K)\) does not necessarily pass to \(C(L)\), where \(L\subset K\). The paper is an extraordinary piece of delicate technical work.
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 7078821
zbMATH Keywords: Markushevich basis
zbMATH Keywords: weakly compactly generated Banach space
zbMATH Keywords: Hilbert generated Banach space
reviewed by: V. Montesinos
MaRDI profile type: Publication
full work available at URL: j.aim.2019.05.012
OpenAlex ID: W2944935977
cites work: Connection Between Strong and Norming Markushevich Bases in Non‐Separable Banach Spaces
cites work: The structure of weakly compact sets in Banach spaces
cites work: On weakly Lindelöf Banach spaces
cites work: Functional-analytic properties of Corson-compact spaces
cites work: The unit ball of the Hilbert space in its weak topology
cites work: Q4895444
cites work: Continuous images of weakly compact subsets of Banach spaces
cites work: Factoring weakly compact operators
cites work: Q3134552
cites work: Q3994935
cites work: Q4339230
cites work: Hilbert-generated spaces.
cites work: The structure of uniformly Gâteaux smooth Banach spaces
cites work: Q3399444
cites work: Banach space theory. The basis for linear and nonlinear analysis
cites work: Asplund spaces and decomposable nonseparable Banach spaces
cites work: Open Problems in the Geometry and Analysis of Banach Spaces
cites work: Biorthogonal systems in Banach spaces
cites work: Some remarks on weakly compactly generated Banach spaces
cites work: Q4042133
cites work: Q4130532
cites work: M-bases in spaces of continuous functions on ordinals
cites work: Q4015700
cites work: Q5748419
cites work: Q3312706
cites work: Q4052703
cites work: Espaces de Banachs faiblement K-analytiques
cites work: Projective resolution of identity in C(K) spaces
cites work: On certain classes of Markushevich bases
cites work: Extensions of Markuševič bases
cites work: Markuševič Bases and Corson Compacta in Duality
cites work: Q4440818
Wikidata QID: Q127825279
DOI: 10.1016/J.AIM.2019.05.012
description: scientific article; zbMATH DE number 7078821