Open sets with the Runge property in Banach spaces
type: Item
instance of: scholarly article
title: Open sets with the Runge property in Banach spaces
zbMATH Open document ID: 1222.46031
author: Jorge Mujica
author: Aaron Zerhusen
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-02-22T00:00:00Z
review text: Let \(E\) and \(F\) be complex Banach spaces. For an open subset \(U\) of \(E\), consider the vector space \({\mathcal H}(U;F)\) of all \(F\)-valued holomorphic mappings on \(U\) equipped with Nachbin's ported topology \(\tau _\omega \). If \(K\) is a compact subset of \(E\), then \({\mathcal H}(K;F)\) denotes the vector space of all \(F\)-valued holomorphic germs on \(K\). The topology \(\tau _\omega \) is defined by the locally convex inductive limit \[ ({\mathcal H}(K;F),\tau _\omega) = \mathop{\text{ind}}\limits_{U \supset K}({\mathcal H}(U;F),\tau _\omega), \] where \(U\) is open in \(E\). In [Topology on spaces of holomorphic mappings, Berlin-Heidelberg-New York: Springer-Verlag (1969; Zbl 0172.39902)], \textit{L. Nachbin} conjectured that the space \(({\mathcal H}(U;F),\tau _\omega)\) coincides with the locally convex projective limit of the spaces \(({\mathcal H}(K;F),\tau _\omega)\) with \(K\subset U\) compact. In 1971, \textit{S. B. Chae} [Ann. Inst. Fourier 21, No.~3, 107--141 (1971; Zbl 0222.46018)] proved Nachbin's conjecture under the hypothesis that the open set \(U\) has the \(F\)-valued Runge property. This means that each compact subset \(K\) of \(U\) is contained in some compact subset \({\tilde K}\) of \(U\) with the property that \({\mathcal H}(U;F)\) is sequentially dense in \(({\mathcal H}({\tilde K};F),\tau _\omega)\). \textit{M. Schottenloher} [Infin. Dim. Holom. Appl., Proc. int. Symp., Univ. Estadual de Campinas/Brasil 1975, 379--391 (1977; Zbl 0397.46044)] showed that every pseudoconvex open subset of a finite-dimensional Banach space has the \({\mathbb C}\)-valued Runge property. The first author has extended Schottenloher's result to the case of pseudoconvex open subsets of Fréchet-Schwartz spaces with a Schauder basis in 1984 and to the case of pseudoconvex Riemann domains over Fréchet-Schwartz spaces with a Schauder basis in 1985. In this paper, the authors improve Schottenloher's result by proving that every pseudoconvex open subset …
reviewed by: Hans-Andreas Braunß
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46A13
zbMATH DE Number: 5854887
zbMATH Keywords: holomorphic mapping
zbMATH Keywords: Banach space
zbMATH Keywords: unconditional Schauder basis
zbMATH Keywords: pseudoconvex open set
zbMATH Keywords: Runge property
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.11.016
OpenAlex ID: W2087673797
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DOI: 10.1016/J.JMAA.2010.11.016
description: scientific article; zbMATH DE number 5854887