Normal structure in James spaces
type: Item
instance of: scholarly article
title: Normal structure in James spaces
zbMATH Open document ID: 0883.46014
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-07-03T00:00:00Z
review text: Let \(X\) be a Banach space with a basis \((x_n)\). Let \(c_0\) be the space of all real sequences that converges to \(0\). The famous James space \(J(X)\) [\textit{R. C. James}, Proc. Natl. Acad. Sci. USA 37, 174-177 (1951; Zbl 0042.36102)] consists of all sequences \((\alpha_n)\in c_0\) for which \[ \sup\Biggl\{\Biggl|\sum_{1\leq i\leq n}(\alpha_{p_i}- \alpha_{p_{i+1}}) x_i+(\alpha_{p_{n+1}}- \alpha_{p_1}) x_{n+1}\Biggr|\Biggr\}<\infty, \] the supremum being taken over all finite increasing sequences of positive numbers \(p_1,p_2,\dots, p_{n+1} \). In this interesting paper, the authors have proved the following important result about the normal structure property of James space \(J(X)\). Main Theorem: Let \(X\) be a Banach space with a basis which is symmetric, boundedly complete, 1-unconditional, and uniformly monotone. Then \(J(X)\) has the normal structure property.
reviewed by: Ismat Beg
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 1005598
zbMATH Keywords: James space
zbMATH Keywords: normal structure
zbMATH Keywords: Banach space with a basis
zbMATH Keywords: symmetric
zbMATH Keywords: boundedly complete
zbMATH Keywords: 1-unconditional
zbMATH Keywords: uniformly monotone
author: M. A. Khamsi
author: Srinivasa Swaminathan
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0466
OpenAlex ID: W2087675761
DOI: 10.1006/JMAA.1996.0466
description: scientific article; zbMATH DE number 1005598