Simultaneous smoothing and interpolation with respect to E. Borel's theorem
type: Item
instance of: scholarly article
title: Simultaneous smoothing and interpolation with respect to E. Borel's theorem
zbMATH Open document ID: 0781.46002
DOI: 10.1007/BF01207462
author: Markus Poppenberg
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: The following tame splitting theorem for short exact sequences of Fréchet spaces is proved: Every tamely exact sequence of the form \[ 0 \longrightarrow E \longrightarrow G\longrightarrow \Lambda^ 1_ \infty(\alpha) \longrightarrow 0 \] splits tamely if \(E\) satisfies a condition called \((\Omega DS)\) which is slightly stronger than property \((\Omega)\) in standard form introduced by D. Vogt und M. J. Wagner; here \(\Lambda^ 1_ \infty(\alpha)\) denotes a power series space of infinite type. In contrast to known results, the above theorem needs neither nuclearity nor the assumption that the seminorms are induced by semiscalar products. The proof is based on a result which combines a classical theorem of E. Borel with the technique of smoothing a function by convolution; this result is used for a verification of a technical condition due to D. Vogt which is sufficient for a time version of \(\text{Proj}^ 1 {\mathcal X}=0\) in the sense of the projective limit functor of V. P. Palamodov.
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 446279
zbMATH Keywords: tame splitting theorem for short exact sequences of Fréchet spaces
zbMATH Keywords: power series space of infinite type
zbMATH Keywords: smoothing a function by convolution
zbMATH Keywords: projective limit functor
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 446279