On rearrangement theorems in Banach spaces
type: Item
instance of: scholarly article
title: On rearrangement theorems in Banach spaces
zbMATH Open document ID: 1297.46012
author: Sergei Chobanyan
author: G. Giorgobiani
author: Vakhtang V. Kvaratskhelia
author: Shlomo Levental
author: V. Tarieladze
published in: Georgian Mathematical Journal
publication date: 2014-06-20T00:00:00Z
review text: A well-known consequence of the famous Steinitz lemma says that, if a subsequence of partial sums of a series \(\sum x_n\) in \(\mathbb R^n\) converges to \(x\) and \(x_n \to 0\) as \(n \to \infty\), then, in some rearrangement of terms, the series converges to \(x\). The paper under review demonstrates that in every infinite-dimensional Banach space \(X\) this result fails drastically. Namely, it is shown that \(X\) contains a series \(\sum x_n\) with the following properties: (a) some subsequence of its partial sums converges in \(X\) and all its partial sums \(\sum_{k=1}^n x_k\) are uniformly bounded; (b) \(\sum_{k=1}^\infty\|x_k\|^p<\infty\) for every \(p\in(2,+\infty)\); (c) for any permutation \(\pi:\mathbb{N}\to\mathbb{N}\) and any sequence \((\theta_n)_{n\in\mathbb{N}}\) with \(\theta_n\in\{-1,1\}\), \(n=1,2,\dots\), the series \(\sum_{k=1}^\infty\theta_kx_{\pi(k)}\) diverges in \(X\). Remark that in some spaces (for example, in \(L_2\)) the condition (b) cannot be substituted with \(\sum_{k=1}^\infty\|x_k\|^2<\infty\) [\textit{M. I. Kadets}, Usp. Mat. Nauk 9, No. 1(59), 107--109 (1954; Zbl 0055.29003)].
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 40A30
Mathematics Subject Classification ID: 40J05
zbMATH DE Number: 6305829
zbMATH Keywords: rearrangements of series
zbMATH Keywords: Banach space
zbMATH Keywords: Sylvester matrix
zbMATH Keywords: Steinitz theorem
zbMATH Keywords: Dvoretzky-Hanani theorem
MaRDI profile type: Publication
full work available at URL: gmj-2014-0016
OpenAlex ID: W2320893744
DOI: 10.1515/GMJ-2014-0016
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description: scientific article; zbMATH DE number 6305829