Peetre's theorem in the locally convex setting
type: Item
instance of: scholarly article
title: Peetre's theorem in the locally convex setting
zbMATH Open document ID: 1332.46033
author: Rafael Dahmen
published in: Forum Mathematicum
publication date: 2015-09-08T00:00:00Z
review text: \textit{J. Peetre}'s theorem [Math. Scand. 7, 211--218 (1959; Zbl 0089.32502)] characterizes linear partial differential operators as local linear maps \(T\) between spaces of \(C^\infty\) functions without mentioning any sort of derivatives. It became a relevant tool in several fields of mathematics. In the paper under review, the author generalizes this theorem in several ways for operators between spaces of \(C^r\) functions defined on a finite-dimensional domain with values in infinite-dimensional locally convex spaces. For general range spaces, the continuity of the operator \(T\) must be assumed. On the other hand, it is proved that local linear operators are necessarily continuous if the smooth functions of class \(r\) take values in Fréchet spaces. A variant of Peetre's theorem for continuous functions defined on a metrizable space without isolated points is also presented.
reviewed by: José Bonet
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 53B99
Mathematics Subject Classification ID: 47F99
zbMATH DE Number: 6479273
zbMATH Keywords: differential operators
zbMATH Keywords: infinite-dimensional differential calculus
zbMATH Keywords: locally convex spaces
zbMATH Keywords: Fréchet spaces
zbMATH Keywords: Peetre's theorem
MaRDI profile type: Publication
full work available at URL: forum-2013-0159
OpenAlex ID: W2314146636
DOI: 10.1515/FORUM-2013-0159
description: scientific article; zbMATH DE number 6479273