Hahn spaces in Fréchet spaces and applications to real sequence spaces
type: Item
instance of: scholarly article
title: Hahn spaces in Fréchet spaces and applications to real sequence spaces
zbMATH Open document ID: 1308.46005
author: Johann Boos
author: Lothar Komp
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2014-09-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1403811849
review text: To investigate the extent to which a sequence space is determined by the sequences of \(0\)'s and \(1\)'s that it contains, \textit{G. Bennett}, the first author and the reviewer in [Stud. Math. 149, No. 1, 75--99 (2002; Zbl 0995.46010); addendum ibid. 171, No. 3, 305--309 (2005; \url{doi:10.4064/sm171--3-7})] introduced the notion of Hahn property. As defined by them, a sequence space \(E\) is said to be a Hahn space if, for each FK-space \(F\), \[ \chi\cap E\subset F\Rightarrow E\subset F, \] where \(\chi\) is the set of all sequences of \(0\)'s and \(1\)'s. In the paper under review, this problem is studied in the more general context of FH-spaces. Let \(H\) be a Fréchet space. A Fréchet space \(E\) is called FH-space if \(E\subset H\) and the inclusion map \(i:E\rightarrow H\) is continuous. Hence, FK-spaces are FH-spaces, where \(H=\omega\), the space of all scalar sequences. A tuple \(( H,\psi) \) with \(\psi\subset H\) is called a Hahn tuple if \(\widehat{\psi}\), the intersection of all FH-spaces containing \(\psi\), is an FH-space. (Note that in the prototypical situation with \(H=\omega\) and \(\psi=\chi\) we have \(\widehat{\chi}=\ell_{\infty}\), the FK-space of all bounded sequences.) A subspace \(E\subset H\) is said to be a Hahn space relative to the Hahn tuple \(( H,\psi)\) if the implication \(\psi\cap E\subset F\Rightarrow E\subset F\) holds for each FH-space \(F\). A Hahn space \(E\) is called big if each subspace \(X\) of \(H\) with \(E\subset X\subset\widehat{\psi}\) is a Hahn space\(.\;\) The authors discuss properties of Hahn tuples and Hahn spaces. It is shown that the famous functional analytical characterization of the Hahn property, which holds in FK-spaces, remains valid for FH-spaces: An FH-space \(E\) is a Hahn space relative to the Hahn tuple \(( H,\psi) \) if and only if \(\langle \psi\cap E\rangle \), the linear span of \(\psi\cap E\), is dense and barrelled in \(E\). For a subspace \(E\) of \(H\), the authors define the lower \(\psi\…
reviewed by: Toivo Leiger
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 40H05
Mathematics Subject Classification ID: 46A08
zbMATH DE Number: 6341730
zbMATH Keywords: Hahn spaces
zbMATH Keywords: FH-spaces
zbMATH Keywords: inclusion theorems
zbMATH Keywords: barrelledness
MaRDI profile type: Publication
OpenAlex ID: W1515791465
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DOI: 10.7169/FACM/2014.50.2.10
description: scientific article; zbMATH DE number 6341730