The closed graph theorem for nonlinear maps
type: Item
instance of: scholarly article
title: The closed graph theorem for nonlinear maps
zbMATH Open document ID: 0929.47032
author: Antonio Tineo
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-12-03T00:00:00Z
review text: The closed graph theorem alluded to in the title can be stated as follows: Let \(X\) be a real vector space with a translation-invariant complete metric \(d\) satisfying \(d(\frac 1n x,0)\to 0\) as \(n\to\infty\) for all \(x\in X\) and \(d(\frac 1m x_n,0)\to 0\) as \(n\to\infty\) whenever \(m\) is a positive integer and \(d(x_n,0)\to 0\). Let \((Y,\rho)\) be a complete metric space and \(f:X\to Y\) a function with closed graph satisfying \(\lim_{n\to\infty}f(x+\frac 1n u)=f(x)\) uniformly in \(x\in X\) whenever \(u\in X\). Then \(f\) is uniformly continuous.
reviewed by: Christian Fenske
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 1289948
zbMATH Keywords: closed graph theorem
zbMATH Keywords: invariant metric
zbMATH Keywords: uniform continuity
MaRDI profile type: Publication
cites work: Q4340096
cites work: Q3249198
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description: scientific article; zbMATH DE number 1289948