Tsirelson superspaces and \(\ell _ p\)
type: Item
instance of: scholarly article
title: Tsirelson superspaces and \(\ell _ p\)
zbMATH Open document ID: 0609.46013
DOI: 10.1016/0022-1236(86)90089-3
author: Steven F. Bellenot
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: Tsirelson space, T, is a Banach space which contains neither \(c_ 0\) nor \(\ell_ p\)-subspaces. Indeed, T contains no subsymmetric sequence. The main parameter in the construction of T, is the definition of what is an ''admissible'' sequence of finite subsets of N, \((E_ i)^ k_ 1\). In T, the \((E_ i)^ k_ 1\) are required to satisfy min \(E_{i+1}>\max E_ i\) and \(\min (\cup E_ i)\geq k.\) By modifying this ''admissible'' to require \(f(\min (\cup E_ i))\geq k\), for other non-decreasing integer functions f, we obtain a large collection of spaces which include the \(\ell_ p\)-spaces and Tsirelson superspaces S. For these spaces S the ''strict'' and ''modified'' norms are equivalent (2- equivalent for T). Some of these spaces S are not isomorphic to the Cartesian square. Thus techniques different from which have been used to study T are developd. Norming trees turn out to be the main tool. For each S, there is a p so that S satisfies an upper \(\ell_ p\)-estimate and lower \(\ell r\)-estimates for each \(r>p\).
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3985914
zbMATH Keywords: equivalent bases
zbMATH Keywords: Tsirelson superspaces
zbMATH Keywords: Cartesian square
zbMATH Keywords: Norming trees
zbMATH Keywords: \(\ell r\)-estimates
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90089-3
OpenAlex ID: W2031100932
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description: scientific article; zbMATH DE number 3985914