Linear mappings are fairly well-behaved
type: Item
instance of: scholarly article
title: Linear mappings are fairly well-behaved
zbMATH Open document ID: 0715.46053
DOI: 10.1007/BF01188684
author: Douglas Bridges
author: Hajime Ishihara
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Classically, it is absurdly trivial that if an element x of a normed space E is not in the kernel of a linear mapping T, then Tx\(\neq 0\). Constructively, however, a stronger, and affirmative, meaning is given to the conclusion Tx\(\neq 0\), and it thereby becomes difficult to prove. In the constructive development of normed space the condition Tx\(\neq 0\) is taken to mean that \(\| Tx\| >0\), where the notion of positive real number is used in the strong constructive sense of explicit positive rational approximations. Typically, the hypotheses are also strengthened accordingly. The main result of the paper (expressed in the title) is that if T maps E onto a Banach space, and if \(\| x-y\| >0\) whenever \(y\in \ker (T)\), then Tx\(\neq 0\). Other results involve the graph of T, with applications to the boundedness of mappings onto finite-dimensional spaces.
reviewed by: Mark Mandelkern
Mathematics Subject Classification ID: 46S30
Mathematics Subject Classification ID: 47S30
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 4177919
zbMATH Keywords: kernel
zbMATH Keywords: graph
zbMATH Keywords: boundedness of mappings
MaRDI profile type: Publication
cites work: Q3803111
cites work: Q3754620
cites work: Q3667432
description: scientific article; zbMATH DE number 4177919