The strict topology in certain weighted spaces of functions
type: Item
instance of: scholarly article
title: The strict topology in certain weighted spaces of functions
zbMATH Open document ID: 0615.46026
DOI: 10.1007/BF01158000
author: V. V. Napalkov
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with abstract versions of theorems which compare various standard topologies on spaces of analytic functions. It continues previous work by the author [Sov. Math. Dokl. 25, 755-758 (1982; Zbl 0551.32003); translation from Dokl. Akad. Nauk SSSR 264, 827-830 (Russian) (1982)] on spaces of entire functions and, in a broader setting, work by \textit{K. D. Bierstedt}, \textit{R. Meise}, and \textit{W. H. Summers} [Trans. Am. Math. Soc. 272, 107-160 (1982; Zbl 0599.46026)], as well as others. The abstract setting consists of a non-empty set X (interpretation: open set in \({\mathbb{C}}^ n)\), an exhaustion of X by an increasing sequence of subsets \(X_ m\) (interpretation: \(X_ m\) compact, \(X_ m\subset int X_{m+1}\), \(\cup X_ m=X)\), the space \(F(X)\) of all complex-valued functions on X, and a linear subspace \(A(X)\) of \(F(X)\) (interpretation: space of analytic functions on X). Particularly important subspaces of \(F(X)\) are \(F_ 0(X)\), consisting of the bounded f such that for each \(\epsilon >0\), \(| f| <\epsilon\) off \(X_ m\) for some m (interpretation: functions that vanish at infinity), and \(F_ c(X)\), consisting of the bounded f that vanish off some \(X_ m\) (interpretation: functions with compact support). Topologies are introduced by means of systems of weights; we shall not define ''weight'' and ''system'' here. To a system of weights V is associated a linear subspace \(FV(X)\) of \(F(X)\), consisting of those f such that \(\| f\|_ v=\sup \{v\cdot | f| \}\) is finite for each \(v\in V\), and \(AV(X)=A(X)\cap FV(X)\); the seminorms \(\| \cdot \|_ v\) make these into locally convex spaces, and certain natural continuity conditions hold. If \({\mathcal V}=\{V_ m\}\) is a sequence of systems of weights, both a new system of weights \(\bar V\) and a new sequence of systems of weights \({\mathcal W}=\{W_ m\}\) associated to \({\mathcal V}\) are defined, and under appropriate naturality conditions the inductive limits \({\mathcal V}A(X)=\li…
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 3996390
zbMATH Keywords: strict topology
zbMATH Keywords: weighted spaces
zbMATH Keywords: spaces of analytic functions
zbMATH Keywords: systems of weights
zbMATH Keywords: inductive limits
zbMATH Keywords: gDF
reviewed by: Stuart J. Sidney
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01158000
OpenAlex ID: W2026654604
description: scientific article; zbMATH DE number 3996390