Normal families of holomorphic functions on infinite-dimensional spaces
type: Item
instance of: scholarly article
title: Normal families of holomorphic functions on infinite-dimensional spaces
zbMATH Open document ID: 1124.46025
author: Paula Takatsuka
published in: Portugaliae Mathematica. Nova Série
publication date: 2006-12-11T00:00:00Z
full work available at URL: https://eudml.org/doc/53552
review text: The article is based on the author's doctoral dissertation with J.\ Mujica. Its aim is to extend some classical results of Montel, Hurwitz and Schottky from holomorphic functions in one (or a finite number of) variable(s) to the infinite-dimensional case. Let \(U\) denote a nonvoid open subset of a complex locally convex space \(E\), and let \({\mathcal H}(U)\) be the space of all complex-valued holomorphic functions on \(U\), endowed with the topology of uniform convergence on the compact subsets of \(U\). In Section 2, it is proved that every locally bounded family \({\mathcal F} \subset {\mathcal H}(U)\) is normal if \(E\) is separable. The separability hypothesis is necessary here. Some related results are also given. Section 3 is devoted to the study of normal families of holomorphic functions with exceptional values. It is proved that, for a separable locally convex space \(E\) and for a connected open subset \(U\) of \(E\), a family \({\mathcal F}\) of holomorphic functions on \(U\) with two different exceptional values \(a\) and \(b\) (i.e., the functions in \({\mathcal F}\) omit the values \(a\) and \(b\)) is \(\mathbb{C}_\infty\)-normal (i.e., each sequence in \({\mathcal F}\) either admits a subsequence converging in \({\mathcal H}(U)\) with respect to the compact-open topology or admits a subsequence which diverges to infinity uniformly on each compact subset of \(U\)). An extension of the Little Picard Theorem follows: If \(f\) is an entire function on a locally convex space which is not constant, then \(f\) attains each complex number, with one possible exception.
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 32A19
Mathematics Subject Classification ID: 32H25
zbMATH DE Number: 5079778
zbMATH Keywords: normal family
zbMATH Keywords: holomorphic functions with exceptional values
zbMATH Keywords: Montel's theorem
zbMATH Keywords: theorem of Hurwitz
zbMATH Keywords: Schottky's theorem
zbMATH Keywords: little Picard theorem
zbMATH Keywords: separability
reviewed by: Klaus Dieter Bierstedt
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5079778