A tame splitting theorem for exact sequences of Fréchet spaces
type: Item
instance of: scholarly article
title: A tame splitting theorem for exact sequences of Fréchet spaces
zbMATH Open document ID: 0823.46002
DOI: 10.1007/BF02572355
author: Markus Poppenberg
author: Dietmar Vogt
published in: Mathematische Zeitschrift
publication date: 1995-11-05T00:00:00Z
full work available at URL: https://eudml.org/doc/174763
review text: The following theorem is proved: Let \(F\), \(G\), \(E\) be Fréchet spaces which are equipped with fundamental systems of Hilbertian seminorms \(\|\;\|_ 0\leq \|\;\|_ 1\leq\dots\) Assume that there are an integer \(p_ 0\), real numbers \(0< \theta_ k< 1\), \(0< \tau_ k< 1\) with \(\theta_ k\geq \tau_{k+ p}\) for all \(p\geq p_ 0\) and all \(k\), and constants \(c_ k> 0\) such that (1) \(\|\;\|^*_ k\leq c_ k\|\;\|^{* 1-\theta_ k}_{k- 1} \|\;\|^{*\theta_ k}_{k+ 1}\) on \((F, \|\;\|_{k- 1})'\), (2) \(\|\;\|_ k\leq c_ k\|\;\|^{1- \tau_ k}_{k- 1} \|\;\|^{\tau_ k}_{k+ 1}\) on \(E\) for all \(k\). Then every tamely exact sequence \[ 0\longrightarrow F\longrightarrow G\longrightarrow E\longrightarrow 0 \] splits tamely. Here \(\|\;\|^*_ k\) denotes the dual norm \(\|\phi \|^*_ k= \sup\{| \phi(x)|: \| x\|_ k\leq 1\}\), \(\phi\in F'\), and a linear map \(T: E\to F\) is called tame if \(\| Tx\|_ n\leq c_ n\| x\|_{n+ b}\) for a fixed \(b\) and all \(n\). As an application we give a short proof of a result of Mityagin and Heinkin on the existence of continuous linear extension operators for holomorphic functions on complex manifolds in strictly pseudoconvex regions in \(\mathbb{C}^ N\).
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 784105
zbMATH Keywords: tame
zbMATH Keywords: splitting
zbMATH Keywords: right inverse
zbMATH Keywords: Nash-Moser
zbMATH Keywords: Fréchet spaces
zbMATH Keywords: fundamental systems of Hilbertian seminorms
zbMATH Keywords: tamely exact sequence
zbMATH Keywords: dual norm
zbMATH Keywords: existence of continuous linear extension operators
zbMATH Keywords: holomorphic functions on complex manifolds in strictly pseudoconvex regions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784105