Holomorphic functions on Fréchet-Montel spaces
type: Item
instance of: scholarly article
title: Holomorphic functions on Fréchet-Montel spaces
zbMATH Open document ID: 0849.46031
DOI: 10.1016/0022-247X(92)90268-I
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: Let \(E\) denote a locally convex space over the field of complex numbers \(\mathbb{C}\) and let \(U\) denote an open subset of \(E\). We let \(H(U)\) denote the space of \(\mathbb{C}\)-valued holomorphic functions on \(U\) and \(\tau_0\) is the compact open topology on \(H(U)\). A semi-norm \(p\) on \(H(U)\) is said to be ported by the compact subset \(K\) of \(U\) if for each open set \(V\), \(K\subset V\subset U\), there exists \(c(V)>0\) such that \(p(f)\leq c(V)|f|_V\) for all \(f\) in \(H(U)\). The locally convex topology on \(H(U)\) generated by the semi-norms ported by compact subsets of \(U\) is denoted by \(\tau_w\). Clearly, \(\tau_0\leq \tau_w\) and a good deal of research has been devoted to finding conditions under which we have \(\tau_0= \tau_w\). In this article we consider this problem for certain Fréchet spaces. In this situation the equality \(\tau_0= \tau_w\) on \(H(U)\) implies that \(U\) is an open subset of a Fréchet-Montel space and so, at the outset, we are confined to such spaces. \dots Our main result is the following Theorem 1. If \(E\) is a Fréchet-Montel space with an unconditional basis of type \((T)\) and \(U\) is a balanced domain in \(E\) then \(\tau_0= \tau_w\) on \(H(U)\).
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A35
zbMATH DE Number: 62789
zbMATH Keywords: holomorphic functions
zbMATH Keywords: compact open topology
zbMATH Keywords: semi-norms ported by compact subsets
zbMATH Keywords: Fréchet-Montel space
zbMATH Keywords: unconditional basis of type \((T)\)
author: Seán Dineen
MaRDI profile type: Publication
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full work available at URL: 0022-247x(92)90268-i
OpenAlex ID: W1988257876
description: scientific article; zbMATH DE number 62789