The regularity of spaces of germs of \(f\)-valued holomorphic functions
type: Item
instance of: scholarly article
title: The regularity of spaces of germs of \(f\)-valued holomorphic functions
zbMATH Open document ID: 0921.46033
author: Nguyen Van Hao
author: Bui Dac Tac
published in: Vietnam Journal of Mathematics
publication date: 1999-09-30T00:00:00Z
review text: It is known for a long time that the space \(H(K)\) of scalar-valued holomorphic germs on compact subsets \(K\) of a Banach space \(E\) is always regular. In the paper under review the regularity of spaces \(H(K,F)\) of holomorphic germs on compact subets of an locally convex space \(E\) with values in a Fréchet space \(F\) is investigated. It comes out that the properties \((DN)\) and \((LB_\infty)\) of \(F\) play an essential role. These properties have been introduced by \textit{D. Vogt} [Arch. Math. 45, 255-266 (1985; Zbl 0621.46001)]. As main results the following theorems are proven. Theorem A. If \(F\) is a reflexive Fréchet space with property \((LB_\infty)\) and \(E\) a quotient space of the power series space of infinite type, then \(H(K,F)\) is regular for any unique compact \(K\) in \(E\). Theorem B. A Fréchet space \(F\) has \((DN)\) if and only if \(H(K,F)\) is regular for all compact subsets \(K\) of \(\mathbb{C}^{\mathbb{N}}\). The proof of Theorem B is based on the following lemma which seems to be of interest for itself. Lemma. Let \(F\) be a Fréchet space with a continuous norm and suppose that \(H(0,F)\) for \(0\in \mathbb{C}^{\mathbb{N}}\) is regular. Then \(H(K,F)\) is regular for all compact subets \(K\) of \(\mathbb{C}^{\mathbb{N}}\).
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A13
zbMATH DE Number: 1239972
zbMATH Keywords: inductive limits
zbMATH Keywords: \((DN)\)-property
zbMATH Keywords: \((LB_\infty)\)-property
zbMATH Keywords: holomorphic germs on compact subsets
zbMATH Keywords: Fréchet space
reviewed by: Heinz Junek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1239972