A projective description of the Nachbin-ported topology
type: Item
instance of: scholarly article
title: A projective description of the Nachbin-ported topology
zbMATH Open document ID: 0876.46031
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-11-25T00:00:00Z
review text: The authors give a projective description of the Nachbin-ported topology \(\tau_\omega\) on \({\mathcal H}(U,X)\), the space of holomorphic mappings on a balanced open subset \(U\) of a Fréchet space \(E\) with values in a Banach space \(X\), provided that \(E\) satisfies some general conditions connected with the BB-property. This generalizes and unifies particular results of \textit{S. Dineen} [Ir. Math. Soc. Bull. 30, 68-69 (1993; Zbl 0799.46047)] for spaces \(E\) admitting a T-Schauder decomposition and of \textit{A. Defant} and \textit{M. Maestre} [Proc. Cambr. Phil. Soc. 115, No. 2, 305-313 (1993; Zbl 0847.46021)] for FM-spaces \(E\) satisfying a BB-property. For this aim the authors use the projective topology \(\tau_\delta\) on \({\mathcal H}(U,X)\) of the mentioned above paper of S. Dineen which is weaker than \(\tau_\omega\), and then they were able to give sufficient and necessary conditions on \(E\) such that \(\tau_\delta= \tau_\omega\) holds true. Recall that a Fréchet space \(E\) is an (FBa)-space, if the bounded subsets of \(E\widehat{\otimes}_\pi X\) split for all Banach spaces \(X\). The key result now is the following: Suppose that \(E_b'\) satisfies a certain countability condition (Heinrich's density condition). Then we have \(\tau_\delta= \tau_\omega\) on \({\mathcal H}(U,X)\) for each Banach space \(X\) complemented in \(X''\) if and only if \(E\in (\text{FBa})\). It is shown that for FM- and FS-spaces \(E\) the (FBa)-condition on \(E\) is equivalent to \(E_b'\in (\text{DFop})\) and \(E\in (\text{QNo})\), respectively, where these conditions stand for operator versions of the DF-property and of the quasinormability.
reviewed by: Heinz Junek
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46M40
zbMATH DE Number: 902464
zbMATH Keywords: DFop-spaces
zbMATH Keywords: QNo-spaces
zbMATH Keywords: Heinrich's density condition
zbMATH Keywords: projective description
zbMATH Keywords: Nachbin-ported topology
zbMATH Keywords: space of holomorphic mappings
zbMATH Keywords: Fréchet space
zbMATH Keywords: BB-property
zbMATH Keywords: T-Schauder decomposition
zbMATH Keywords: FM-spaces
zbMATH Keywords: FS-spaces
zbMATH Keywords: (FBa)-condition
zbMATH Keywords: DF-property
zbMATH Keywords: quasinormability
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0044
OpenAlex ID: W2047517327
DOI: 10.1006/JMAA.1996.0044
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description: scientific article; zbMATH DE number 902464