Diffeomorphism invariant norms on \(C^k(X)\)
type: Item
instance of: scholarly article
title: Diffeomorphism invariant norms on \(C^k(X)\)
zbMATH Open document ID: 0935.58007
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1999-12-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73456
review text: The main result of the paper is the following: If \(X\) is a compact manifold without boundary of class \(C^k\) with \(k>0\) and \(\dim(X)>1\), then every \(C^k\)-invariant semi-norm on the space \(C^k(X)\) depends for \(f\in C^k(X)\) only on \(\sup(f)\) and \(\inf(f)\) and is either equivalent to the sup-norm or is a multiple of the semi-norm \(\sup(f) -\inf(f)\). The statement also holds for \(k=0\) if the semi-norm is continuous to start with. The author also obtains results for non-connected manifolds and manifolds with boundary. In an appendix it is shown that \(C_0(S^1)\) is not finitely generated as module over \(\text{Homeo}_0(S^1)\).
Mathematics Subject Classification ID: 58C05
zbMATH DE Number: 1240928
zbMATH Keywords: semi-norms on function spaces
zbMATH Keywords: diffeomorphism invariance
author: Bruno Sevennec
MaRDI profile type: Publication
OpenAlex ID: W2334555780
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DOI: 10.5802/AFST.901
description: scientific article; zbMATH DE number 1240928