Normal structure and the sum-property
type: Item
instance of: scholarly article
title: Normal structure and the sum-property
zbMATH Open document ID: 0629.46016
author: Thomas R. Landes
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The first permanence result for normal structure is due to Belluce, Kirk and Steine: The \(\ell^ N_{\infty}\)-direct sum of N normally structured normed spaces again has normal structure. The author of this paper continues for discussion on it. For \(Z=(R^ N,| \cdot |)\) with symmetric norm \(| \cdot |\) the Z-direct sum of the normed spaces \(X_ 1,...,X_ N\) is their product space with norm \(\| (x_ i)^ N_{i=1}\| =| (\| x_ i\|)^ N_{i=1}|\). A normed space X is said to have the sum-property (SP) if each Z-direct sum of finitely many copies of X has normal structure (NS). The following results are mainly proved in this paper: (1) NS is preserved under the Z-direct-sum-operation iff each line segment (if there is any) in the unit sphere of Z lies in a hyperplane \(\{z|\) \(z_ i=\alpha\}\) for some \(i\leq N\), \(\alpha\neq 0.\) (2) Every infinite dimensional separable normed space can be renormed to have NS and not the SP. All results hold also for weakly NS and weak SP.
reviewed by: Zhang Guanglu
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 4024009
zbMATH Keywords: normal structure
zbMATH Keywords: sum-property
zbMATH Keywords: Z-direct-sum-operation
MaRDI profile type: Publication
full work available at URL: pjm.1986.123.127
OpenAlex ID: W2072283569
DOI: 10.2140/PJM.1986.123.127
description: scientific article; zbMATH DE number 4024009