Topologization of entire sequences spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 0974.46007
author: Yu. Yu. Estinko
publication date: 2001-08-08T00:00:00Z
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A20
zbMATH DE Number: 1631398
zbMATH Keywords: topologization
zbMATH Keywords: entire sequences space
zbMATH Keywords: finite sequence
zbMATH Keywords: locally convex space
MaRDI profile type: Publication
title: Topologization of entire sequences spaces
published in: Naukovyĭ Visnyk Uzhgorods'kogo Universytetu. Seriya Matematyka
review text: This article deals with the locally convex sequence spaces of entire functions. Denote by \(\omega\) the space of all sequences \(x=(x_{n}), n\in\{0,1,2,\ldots\}\), \(x_{n}\) is a complex number. Let \(\varphi\) be a space of all finite sequences. If \(E\) is a subspace of \(\omega\) and \(\varphi\subset E\), then the bilinear form \((x,y)\to\langle x,y\rangle=\sum x_{n}y_{n}, x\in\varphi, y\in E\) defines the dual pair \((\varphi,E)\). By \(\tau(\varphi,E)\) we denote the strongest topology which is consistent with duality \((\varphi,E)\), and by \(\beta (\varphi,E)\) we denote the strongest topology of uniform convergence. The author proves that for any space \(E\) such that \(\varphi\subset E\subset \omega\) the topological space \((\varphi,\beta(\varphi,E))\) coincides with the space \((\varphi,\beta(\varphi,\omega))=(\varphi,\tau(\varphi,\omega))\) and it is a complete space. The topology \(\beta(\varphi,\omega)\) is stronger then \(\sigma(\varphi,E)\). Here \(\sigma(\varphi,E)\) is the weakest topology in which the pre-norm \(x\mapsto|\langle x,y\rangle|\) is continuous for each \(y\in E\). Also the topologies \(\tau(E,F)\), \(\beta(E,F)\) are studied, where spaces \(E, F\) are such that \(\varphi\subset E\subset I\), \(\varphi\subset F\subset E^{*}\), \(I\) is a space of all entire sequences, \(E^{*}\) is a dual space for \(E\).
reviewed by: Aleksandr D. Borisenko
description: scientific article; zbMATH DE number 1631398