Holomorphic functions on locally closed convex sets and projective descriptions
type: Item
instance of: scholarly article
title: Holomorphic functions on locally closed convex sets and projective descriptions
zbMATH Open document ID: 1086.46003
author: José Bonet
author: Sergej N. Melikhov
author: Reinhold Meise
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-11-25T00:00:00Z
review text: A subset \(Q\) of \({\mathbb C}^N\) is locally closed if for each \(z\in Q\) there is a closed neighborhood \(U\) of \(z\) in \({\mathbb C}^N\) such that \(Q\cap U\) is closed. Every open subset and every compact subset of \({\mathbb C}^N\) is locally closed. A convex locally closed set \(Q\) admits an increasing fundamental sequence \((Q_n)_n\) of convex compact sets. Let \({\mathcal V}=(v_{n,k})_{n,k\in\mathbb N}\) be the double sequence of strictly positive upper semicontinuous functions on \({\mathbb C}^N\) given by \(v_{n,k}(z):=\exp (-H_n(z)-| z| /k)\), \(n,k\in{\mathbb N}\), \(z\in {\mathbb C}^N\), where \(H_n\) is the support function of the convex compact set \(Q_n\), \(n\in{\mathbb N}\). The weighted inductive limit of Fréchet spaces \( VH({\mathbb C}^N)\) of entire functions associated with \(V\) is defined by \[ VH({\mathbb C}^N):=\text{ ind}_n\text{ proj}_kH(v_{n,k},{\mathbb C}^N), \] where the steps \(H(v,{\mathbb C}^N)\) are defined, for a positive function \(v\) on \({\mathbb C}^N\), as the Banach space of entire functions \[ H(v,{\mathbb C}^N):=\{f\in H({\mathbb C}^N):\;\| f\| _v:=\sup_{z\in {\mathbb C}^N}v(z)| f(z)| <\infty\}. \] The space \( VH({\mathbb C}^N)\) is a Hausdorff (LF)-space. Its projective hull \(H\bar{V}({\mathbb C}^N)\) is defined by \[ H\bar{V}({\mathbb C}^N):=\{f\in H({\mathbb C}^N):\;\| f\| _{\bar{v}}:=\sup_{z\in {\mathbb C}^N}\bar{v}(z)| f(z)| <\infty \;\forall \bar{v}\in \bar{V}\}, \] where \(\bar{V}\) is the system of all those functions \(\bar{v}\colon {\mathbb C}^N\to [0,\infty)\) which are upper semicontinuous on \({ \mathbb C}^N\) and have the property that for each \( n\) there are \( k=k(n)\) and \(\alpha_n>0\) such that \(\bar{v}\leq \alpha_n v_{n,k}\) on \({\mathbb C}^N\). The projective hull \(H\bar{V}({\mathbb C}^N)\) is a complete locally convex space with respect to the Hausdorff locally convex topology defined by the system of seminorms \(\{\| \;\| _{\bar{v}}:\;\bar{v}\in \bar{V}\}\), and \(VH({\mathbb C}^N)\) is …
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 2118938
zbMATH Keywords: weighted spaces of entire functions
zbMATH Keywords: weighted inductive limits
zbMATH Keywords: projective descriptions
zbMATH Keywords: locally closed sets
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2118938