Generalized prequojections and bounded maps
type: Item
instance of: scholarly article
title: Generalized prequojections and bounded maps
zbMATH Open document ID: 0677.46001
DOI: 10.1007/BF03322453
author: V. B. Moscatelli
author: Giorgio Metafune
published in: Results in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: A locally convex space E is called a generalized prequojection if for each 0-neighborhood U there is another one V such that for every 0- neighborhood W one can find \(c>0\) with \[ W^ 0\cap E'_{U^ 0}\subset cV^ 0. \] The name derives from the fact that a Fréchet space E is a generalized prequojection if and only if it is a prequojection (or, equivalently, does not satisfy condition (*) of Bellenot-Dubinsky). - We recall that a Fréchet space E is a prequojection precisely if its strong bidual \(E''\) is a quojection; i.e., \(E''\) is the strict projective limit of a sequence of Banach spaces. The condition in the definition of generalized prequojection was introduced by \textit{T. Terzioǧlu} [Doǧa T. J. Math. 10, 338-344 (1986)] who also proved that \(L(E,F)=LB(E,F)\) holds if E is a generalized prequojection and F is a barrelled space with a continuous norm and a (Schauder) basis. The present authors first show how it is possible to construct non- trivial generalized prequojections with continuous norm by starting from an arbitrary non-quasi-reflexive Banach space X (using strongly non- norming subspaces of \(X')\). Then they generalize Terzioǧlu's result to the natural limits of its validity by proving that for a generalized prequojection E, the equality L(E,F)\(=LB(E,F)\) remains valid for all locally convex spaces F with bounded approximation property and a continuous norm. [This result was found independently by \textit{A. Galbis} in his dissertation, Valencia (1988).]
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A32
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 4109345
zbMATH Keywords: Schauder basis
zbMATH Keywords: bounded linear maps
zbMATH Keywords: quojection
zbMATH Keywords: barrelled space
zbMATH Keywords: construct non-trivial generalized prequojections with continuous norm
zbMATH Keywords: \(L(E,F)=LB(E,F)\)
zbMATH Keywords: bounded approximation property
reviewed by: Klaus Dieter Bierstedt
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322453
OpenAlex ID: W2146041476
description: scientific article; zbMATH DE number 4109345