Extracting long basic sequences from systems of dispersed vectors
type: Item
instance of: scholarly article
title: Extracting long basic sequences from systems of dispersed vectors
zbMATH Open document ID: 1259.46012
author: Jarno Talponen
published in: Advances in Mathematics
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.0071
review text: The author studies general geometric properties of a non-separable Banach space \(X\) which allow to extract long basic sequences from systems of dispersed vectors. These properties are much weaker than WCG and closely related to Corson's property \((C)\). For example, the space \(X\) has property \((\mathcal{B})\) if, for each nested sequence of closed linear subspaces \((Z_{\alpha})_{\alpha<\kappa}\) of \(X\) such that \(\bigcap_{\alpha<\kappa}Z_{\alpha}=\{0\}\), the subspace \(\bigcup_{\alpha<\kappa}Z_{\alpha}^\bot\subset X^*\) is norming. Given a limit ordinal \(\kappa\), a sequence \((x_{\alpha})_{\alpha<\kappa}\) of \(X\) is strongly dispersed if \([x_\alpha]_{\alpha>\beta} \neq[x_\alpha]_{\alpha>\gamma}\) for \(\beta\neq\gamma\) and \(\bigcap_{\beta<\kappa}[x_\alpha]_{\alpha>\beta}=\{0\}\). One of the author's result says: If \(X\) satisfies \((\mathcal{B})\) and \((x_{\alpha})_{\alpha<\kappa}\) is strongly dispersed, then there is a long basic sequence \((x_{\alpha_\sigma})_{\sigma<\kappa}\).
reviewed by: A. M. Plichko
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46M40
zbMATH DE Number: 6084092
zbMATH Keywords: basic sequence
zbMATH Keywords: transfinite sequence
zbMATH Keywords: minimal sequence
zbMATH Keywords: biorthogonal system
zbMATH Keywords: WCG space
zbMATH Keywords: WLD space
zbMATH Keywords: property \(C\)
zbMATH Keywords: dispersed sequence
zbMATH Keywords: Baire category
zbMATH Keywords: inverse limit
MaRDI profile type: Publication
OpenAlex ID: W1975021824
arXiv ID: 1011.0071
cites work: Unconditional structures of weakly null sequences
cites work: Indecomposability and unconditionality in duality
cites work: Methods in the theory of hereditarily indecomposable Banach spaces
cites work: On bases and unconditional convergence of series in Banach spaces
cites work: Characterization of the Radon-Nikodym Property in terms of inverse limits
cites work: The Weak Topology of a Banach Space
cites work: Unconditional basic sequences in spaces of large density
cites work: Transfinite bases of subspaces in Hausdorff linear topological spaces
cites work: Hierarchies in Recursive Function Theory
cites work: Functional analysis and infinite-dimensional geometry
cites work: A new dichotomy for Banach spaces
cites work: On the Kunen–Shelah properties in Banach spaces
cites work: Biorthogonal systems in Banach spaces
cites work: Valdivia compacta and equivalent norms
cites work: Q4521634
cites work: On a problem of Rolewicz about Banach spaces that admit support sets
cites work: Banach spaces with projectional skeletons
cites work: Set theory. An introduction to independence proofs
cites work: Weak compactness, transfinite Schauder bases and differentiability
cites work: Q3413630
cites work: Q4440808
cites work: Q3709279
cites work: On a question of H.H. Corson and some related problems
cites work: Q3819527
cites work: Q3029573
cites work: A Banach space with few operators
cites work: Uncountable constructions for B. A., e. c. groups and Banach spaces
cites work: Q5596501
cites work: Q3920145
cites work: The Radon-Nikodym Property in Conjugate Banach Spaces
cites work: Lindelöf type of generalization of separability in Banach spaces
cites work: Biorthogonal systems and quotient spaces via Baire category methods
cites work: Some properties of weakly countably determined Banach spaces
cites work: Q5572581
DOI: 10.1016/J.AIM.2012.06.008
description: scientific article; zbMATH DE number 6084092