Splitting of exact sequences of Fréchet spaces in the absence of continuous norms
type: Item
instance of: scholarly article
title: Splitting of exact sequences of Fréchet spaces in the absence of continuous norms
zbMATH Open document ID: 1065.46003
author: Dietmar Vogt
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-12T00:00:00Z
review text: This paper studies the splitting of short exact sequences \[ 0 \rightarrow G \rightarrow E \rightarrow F \rightarrow 0 \tag{1} \] of nuclear Fréchet spaces the in the case that \(F\) does not necessarily have a continuous norm. The most important case is when \(E\) and \(F\) are spaces of \(C^{\infty}\) functions on an open subset \(\Omega\) of \(\mathbb{R}^n\). Valdivia and Vogt proved independently that this space is isomorphic to a countable product of copies of the Fréchet space \(s\) of rapidly decreasing sequences. The splitting theorem of Vogt and Wagner ensures that, if \(G\) has property \((\Omega)\) and \(F\) has property (DN), then the exact sequence (1) splits, i.e., the quotient map \(A:E \rightarrow F\) has a continuous linear right inverse. However, property (DN) implies the existence of a continuous norm. \textit{P. Domański} and \textit{D. Vogt} [J. Funct. Anal. 153, 203--248 (1998; Zbl 0912.46026)] developed a theory of graded Fréchet spaces and graded exact sequences, with important applications to the splitting of differential complexes. In the present paper, the author takes a different approach: instead of imposing conditions on the type of exactness, he assumes restrictions on the maps admitted in the exact sequence. A map is called seminorm-kernel continuous (SK-continuous) if it is continuous for the additive group topologies on \(E\) and \(F\) whose neighbourhoods of the origin are the kernels of the continuous seminorms. The condition that the quotient map \(A:E \rightarrow F\) is an SK-homomorphism is natural, since it is satisfied if the sequence (1) splits. Two new topological invariants \((DN_{loc})\) and \((\Omega_{\text{loc}})\) are introduced and studied. The main result of the article asserts that if the quotient map \(A:E \rightarrow F\) is an SK-homomorphism, \(F\) has property \((DN_{\text{loc}})\) and \(G\) has property \((\Omega_{\text{loc}})\), then the exact sequence (1) splits. This is applied to obtain a structure theory o…
reviewed by: José Bonet
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A63
Mathematics Subject Classification ID: 46M18
Mathematics Subject Classification ID: 46E05
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46N20
zbMATH DE Number: 2107222
zbMATH Keywords: Fréchet spaces
zbMATH Keywords: short exact sequences
zbMATH Keywords: right inverses
zbMATH Keywords: continuous norm
zbMATH Keywords: property (DN)
zbMATH Keywords: property (\(\Omega)\)
zbMATH Keywords: differential complexes
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.05.014
OpenAlex ID: W1963746252
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DOI: 10.1016/J.JMAA.2004.05.014
description: scientific article; zbMATH DE number 2107222