On some dense compact operator range subspaces in separable Fréchet spaces
type: Item
instance of: scholarly article
title: On some dense compact operator range subspaces in separable Fréchet spaces
zbMATH Open document ID: 0733.47021
DOI: 10.1007/BF01903705
published in: Acta Mathematica Hungarica
publication date: 1990-01-01T00:00:00Z
review text: Let E be a (Hausdorff) topological vector space and X, Y two closed subspaces of E with \(X\cap Y=0\). If \(X+Y\) is dense in E (and \(X+Y\neq E)\), then X, Y are called (proper) quasi-complements. With these notations, the author proves the following: If T: \(F\to E\) is a continuous not relatively open linear operator from a Fréchet space F into a separable Fréchet space E with dense range, then for every infinite dimensional separable Banach space Z, there exists a compact injective linear operator Q: \(Z\to E\) such that Q(Z) is dense and \(Q(Z)\cap T(F)=0.\) If X, Y are proper quasi-complements in a separable Fréchet space E, then E contains a dense subspace \(G\supset Y\) with \(G\cap X=0.\) This generalizes a theorem of \textit{V. V. Shevchik} [Dokl. Akad. Nauk SSSR 263, 817-819 (1982; Zbl 0514.46009)].
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 4212817
zbMATH Keywords: quasi-complements
zbMATH Keywords: compact injective linear operator
zbMATH Keywords: quasi-complements in a separable Fréchet space
author: Jerzy Kąkol
reviewed by: D. Somasundaram
MaRDI profile type: Publication
cites work: A solution to a problem of De Wilde and Tsirulnikov
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cites work: On subseries convergent series and m-quasi-bases in topological linear spaces
cites work: Quasi-Complements and Closed Projections in Reflexive Banach Spaces
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description: scientific article; zbMATH DE number 4212817