Sequences of 0's and 1's. Classes of concrete `big' Hahn spaces
type: Item
instance of: scholarly article
title: Sequences of 0's and 1's. Classes of concrete `big' Hahn spaces
zbMATH Open document ID: 1061.46004
DOI: 10.4171/ZAA/1175
author: Johann Boos
author: Maria Zeltser
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-06-15T00:00:00Z
review text: Let \(\omega\) denote the space of all real sequences \(x= (x_k)\) \((k\in\mathbb{N})\). The space \(\omega\), supplied with the topology of coordinatewise convergence, is an FK-space, i.e., a sequence space endowed with a complete, metrizable l.c. topology such that all the mappings \(x\to x_k\) \((k\in\mathbb{N})\) are continuous. A normable FK-space is called a BK-space. Any vector subspace of \(\omega\) is called a sequence space. A sequence space can have at most one FK-topology. Let \(\chi\) be the set of all sequences of \(0\)'s and \(1\)'s and, for any sequence space \(E\), let \(\chi(E)\) denote the linear hull of the sequences of \(0\)'s and \(1\)'s contained in \(E\). It is always true that \(\chi(E)\subset\chi(\ell^\infty)\cap E\). For any infinite matrix \(B= (b_{nk})\) with real entries, the summability domain \(c_B\) of \(B\) is defined as \[ c_B= \biggl\{x= (x_k)\in \omega\mid Bx:= \Bigl(\sum_k b_{nk} x_k\Bigr)_n\in c\biggr\} \] (\(Bx\) is assumed to be well-defined). Let \(\lim_B\) denote the mapping \(\lim_B: c_B\to\mathbb{R}\), \(x\mapsto\lim Bx\). Then the pair \((c_B,\lim_B)\) is called a matrix method. \(c_B\) is a separable FK-space and, if the matrix method is conservative, i.e., \(c\subset c_B\), its bounded domain \(\ell^\infty\cap c_B\) is a closed subspace of \(\ell^\infty\). A sequence space \(E\) is said to have the Hahn property if for any FK-space \(F\) we have \[ \chi(E)\subset F\Rightarrow E\subset F. \] In this case, \(E\) is called a Hahn space. If, moreover, \(F\) is a separable FK-space, then \(E\) is said to have the separable Hahn property. Finally, \(E\) has the matrix Hahn property if for a matrix method \((c_B,\lim_B)\) we have \(\chi(E)\subset c_B\Rightarrow E\subset c_B\). It has been proved [\textit{G. Bennett} and \textit{N. J. Kalton}, Can. J. Math. 25, 511--524 (1973; Zbl 0272.46009)] that \(E\) has the Hahn property if and only if \(\chi(E)\) is a dense barrelled subspace of \(E\). Also, \textit{G. Bennett, J. Boos} …
reviewed by: Constantine G. Lascarides
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 40C05
Mathematics Subject Classification ID: 40H05
zbMATH DE Number: 2075232
zbMATH Keywords: dense barrelled subspaces
zbMATH Keywords: Hahn space (property)
zbMATH Keywords: separable Hahn property
zbMATH Keywords: matrix Hahn property
zbMATH Keywords: bounded domains
zbMATH Keywords: weighted mean method
zbMATH Keywords: Nörlund method
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1175
OpenAlex ID: W2076675687
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description: scientific article; zbMATH DE number 2075232