An improvement of a recent closed graph theorem
type: Item
instance of: scholarly article
title: An improvement of a recent closed graph theorem
zbMATH Open document ID: 1163.46002
author: Ronglu Li
author: Sok Yoo Won
author: Shuhui Zhong
published in: Topology and its Applications
publication date: 2008-09-23T00:00:00Z
review text: Let \(X\) and \(Y\) be topological vector spaces. The authors call a mapping \(f:X\to Y\) weakly quasilinear if \(f\) satisfies: (1)~if \(f(x_n)\to 0\) and \(f(u_n)\to0\), then \(f(x_n+u_n)\to0\); (2)~if \(f(x_n-x)\to0\) and \(t_n\to t\) in the scalar field \(K\), then \(f(t_nx_n-tx)\to0\); and (3)~if \(x_n-u\to0\) in \(X\), then \(f(x_n)\to f(u)\) if and only if \(f(x_n-u)\to0\). They show that this property of \(f\) is properly weaker than the property of being quasi-linear in the sense of a paper of the first two authors [J. Math. Anal. Appl. 311, No.~2, 736--743 (2005; Zbl 1087.46003)]. Then, they extend the closed graph theorem from the mentioned paper to the case of weakly quasi-linear mappings.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 47H99
zbMATH DE Number: 5346273
zbMATH Keywords: weakly quasilinear mappings
zbMATH Keywords: quasilinear mappings
zbMATH Keywords: closed graph
zbMATH Keywords: Fréchet spaces
MaRDI profile type: Publication
full work available at URL: j.topol.2008.05.012
OpenAlex ID: W2033203774
cites work: Q4038016
cites work: The closed graph theorem for nonlinear maps
cites work: Ultracomplete convergence vector spaces and the closed graph theorem
cites work: Continuity of mappings between Fréchet spaces
cites work: Q4178057
DOI: 10.1016/J.TOPOL.2008.05.012
description: scientific article; zbMATH DE number 5346273