Complete rank theorem of advanced calculus and singularities of bounded linear operators
type: Item
instance of: scholarly article
title: Complete rank theorem of advanced calculus and singularities of bounded linear operators
zbMATH Open document ID: 1176.47049
author: Jipu Ma
published in: Frontiers of Mathematics in China
publication date: 2008-09-08T00:00:00Z
review text: The article deals with generalizations of classical results in calculus about smooth systems of functions in several variables the local representation of a function \(f: {\mathbb R}^m \to {\mathbb R}^n\), \(f \in C^r\), in the form \[ f(x) = (\psi \circ f'(x_0) \circ \varphi)(x),\tag{1} \] where \(\varphi, \psi(x) \in C^r\), \(\varphi(x_0) = 0\), \(\varphi'(x_0) = I\), \(\psi(0) = f(x_0)\), \(\psi'(0) = I\). The author remarks that this representation is true if \(f \in C^r\) is a mapping between Banach spaces \(E\) and \(F\) and \(\text{ker} \, f'(x_0)\) and \(\text{im} \, f'(x_0)\) are complemented subspaces in \(E\) and \(F\) correspondingly. The proof repeats the standard arguments. In connection with this theorem, the author repeats standard theorems about the stability properties of linear Fredholm and semi-Fredholm operators, some results about the existence of generalized inverse operators, and so forth.
Mathematics Subject Classification ID: 47J07
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 5322575
zbMATH Keywords: rank theorem
zbMATH Keywords: generalized inverse operators
zbMATH Keywords: Fredholm and semi-Fredholm operators
zbMATH Keywords: local linearisation
zbMATH Keywords: perturbation analysis
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: s11464-008-0019-8
OpenAlex ID: W2060980407
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DOI: 10.1007/S11464-008-0019-8
description: scientific article; zbMATH DE number 5322575