\((LB^{\infty})\)-structure of spaces of germs of holomorphic functions
type: Item
instance of: scholarly article
title: \((LB^{\infty})\)-structure of spaces of germs of holomorphic functions
zbMATH Open document ID: 0968.46013
DOI: 10.5565/PUBLMAT_44100_06
author: Nguyen Dinh Lan
published in: Publicacions Matemàtiques
publication date: 2001-09-16T00:00:00Z
full work available at URL: https://eudml.org/doc/41388
review text: For a nuclear Fréchet space \(E\) and a balanced convex compact subset \(B\) of \(E\) denote by \(H(B)\) the space of germs of holomorphic functions on \(B\). Then \(H(B)'\) is a nuclear Fréchet space. Consider the following assertions: (1) \(E\) has the property \((\widetilde\Omega_B)\). (2) \(H(B)'\) has the property \((LB^\infty)\). (3) The set \(B\) is not pluripolar in \(E\). The author proves that (1) implies (2) and that (1)--(3) are equivalent, provided that \(E\) has the bounded approximation property. Under the latter hypothesis the equivalence of (1) and (3) was shown in \textit{S. Dineen}, \textit{R. Meise} and \textit{D. Vogt} [Bull. Soc. Math. Fr. 112, 41-68 (1984; Zbl 0556.46003)]. Ideas from this paper and from \textit{D. Vogt} [J. Reine Angew. Math. 345, 182-200 (1983; Zbl 0261.46003)] are used in the proof. The author also shows that if \(E\) has a Schauder basis, the space \(H(B)'\) satisfies \((\widetilde\Omega_B)\) (resp. \((\overline\Omega_B)\)) if and only if \(E\) has the same property.
reviewed by: Reinhold Meise
Mathematics Subject Classification ID: 46A63
Mathematics Subject Classification ID: 46E50
zbMATH DE Number: 1505970
zbMATH Keywords: germs of holomorphic functions
zbMATH Keywords: linear topological invariants of \(\Omega\)-type
zbMATH Keywords: nuclear Fréchet space
zbMATH Keywords: space of germs of holomorphic functions
MaRDI profile type: Publication
OpenAlex ID: W2116890587
description: scientific article; zbMATH DE number 1505970