Metrizability of spaces of holomorphic functions with the Nachbin topology
type: Item
instance of: scholarly article
title: Metrizability of spaces of holomorphic functions with the Nachbin topology
zbMATH Open document ID: 1136.46033
author: José M. Ansemil
author: Socorro Ponte
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-07-19T00:00:00Z
review text: The authors show that the space of all holomorphic functions on an open subset \(U\) of a complex metrizable locally convex space \(E\), endowed with the Nachbin ported topology \(\tau_\omega\), is metrizable only if \(E\) has finite dimension. This answers a question posed in [\textit{J.\,Mujica}, ``Gérmenes holomorfos y funciones holomorfas en espacios de Fréchet'' (Publicaciones del Departamento de Teoría de Funciones, Universidad de Santiago, Spain) (1978; per bibl.)]. The following proposition is an important step of the proof and is of independent relevance. Let \(E\) be a metrizable locally convex space and let \(\tau\) be a locally convex topology between the compact open topology \(\tau_0\) and \(\tau_\omega\). If \((E',\tau)\) is metrizable, then \(E\) is a normed space.
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46E50
zbMATH DE Number: 5173012
zbMATH Keywords: holomorphic function
zbMATH Keywords: Nachbin topology
zbMATH Keywords: metrisable space
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
full work available at URL: j.jmaa.2007.01.012
OpenAlex ID: W2073139405
cites work: Q4049606
cites work: Q3688914
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cites work: Weak sequential convergence in the dual of a Banach space does not imply norm convergence
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cites work: Spaces of holomorphic mappings on Banach spaces with a Schauder basis
DOI: 10.1016/J.JMAA.2007.01.012
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description: scientific article; zbMATH DE number 5173012