Pointwise convergence of Cauchy sequences
type: Item
instance of: scholarly article
title: Pointwise convergence of Cauchy sequences
zbMATH Open document ID: 0656.46026
DOI: 10.1216/RMJ-1987-17-1-85
author: Paul Creegan
published in: Rocky Mountain Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: If G is a normed space whose elements are real-valued functions on a set X it is sometimes interesting or even important to know whether the completion of G can also be realized as a space of real-valued functions on X. \textit{J. W. Brace} and \textit{J. D. Thomison} used in Ill. J. Math. 22, 581-602 (1978; Zbl 0384.46016) and in Ill. J. Math. 23, 93-100 (1979; Zbl 0391.46022) the notion of \({\mathcal F}\)-topology to study the problem: If \({\mathcal F}\) is a filter on X, then (G,\({\mathcal F})\) is the topological group whose zero-neighbourhood basis is given by \[ \{\{f\in G:\quad there\quad is\quad an\quad F\in {\mathcal F}\quad with\quad f(F)\subseteq [- \epsilon,\epsilon]\},\quad \epsilon >0\}. \] They gave a representation of the desired form for \(G=(G,{\mathcal F})\) if X is linear, \({\mathcal F}\) convex and circled and moreover every convergent sequence in G converges uniformly on some \(F\in {\mathcal F}\). If the condition is relaxed, then additional conditions of another kind are required. In this paper the autor considers Cauchy-sequences instead of convergent sequences and gets similar results.
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 4071660
zbMATH Keywords: pointwise Cauchy filter convergence space
zbMATH Keywords: uniform Cauchy filter convergence space
zbMATH Keywords: subuniform Cauchy filter convergence space
zbMATH Keywords: subpointwise Cauchy filter convergence space
zbMATH Keywords: completion
zbMATH Keywords: Cauchy- sequences
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4071660