Convergence structures in numerical analysis
type: Item
instance of: scholarly article
title: Convergence structures in numerical analysis
zbMATH Open document ID: 0853.54010
DOI: 10.4171/ZAA/713
author: Siegfried Gähler
author: D. Matel-Kaminska
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1997-01-09T00:00:00Z
review text: Summary: The paper deals -- under the viewpoint of topology -- with discrete Cauchy spaces, which are spaces where a discrete Cauchy structure \((t, {\mathcal C})\) (with \(t\) being a discrete convergence and \({\mathcal C}\) being a discrete pre-Cauchy structure) is defined. More precisely, let \(E_1, E_2, \dots\) and \(E\) be arbitrary sets and let \({\mathcal S}\) denote the set of all discrete sequences \((x_n )_{n\in N'}\) with \(x_n\in E_n\) \((n\in N')\) and with \(N'\) being an infinite subset of \(\mathbb{N}= \{1, 2, \dots\}\). Then \(t\) and \({\mathcal C}\) are certain subsets of \(({\mathcal S}, E)\) respectively of \({\mathcal S}\), which in a certain sense are assumed to be compatible. The paper gives properties of \(t\) and \({\mathcal C}\) and among others is devoted to the problem of completion of discrete Cauchy spaces \((((E_1, E_2, \dots), E); (t,{\mathcal C}))\). The construction of a completion of a discrete Cauchy space differs (in some sense essentially) from the construction of a completion of a usual sequential Cauchy space and is even more simple. An essential part of the paper is devoted to certain metric discrete Cauchy spaces, where -- among others assuming that \(E\) is equipped with a metric \(d\) and that there exist mappings \(q_n: E_n\to E\) \((n\in \mathbb{N})\) -- the discrete Cauchy structure \((t, {\mathcal C})\) is defined by \[ \begin{aligned} ((x_n )_{N'}, x)\in t &\iff (d(q_n (x_n), x))_{N'}\to 0\\ (x_n )_{N'}\in {\mathcal C} &\iff (q_n (x_n ))_{N'} \text{ is a Cauchy sequence in } (E,d). \end{aligned} \] It turns out that such a metric discrete Cauchy space is complete if and only if \((E, d)\) is complete and that also the completion is metric. A further subject of the paper are metric discrete Cauchy spaces of mappings between metric discrete Cauchy spaces, where simple characterizations of the corresponding discrete convergence and discrete pre-Cauchy structure of such a discrete Cauchy space as well as a necessary and…
Mathematics Subject Classification ID: 54A99
Mathematics Subject Classification ID: 65J99
Mathematics Subject Classification ID: 54C99
Mathematics Subject Classification ID: 65D99
zbMATH DE Number: 889010
zbMATH Keywords: metric discrete Cauchy spaces
zbMATH Keywords: discrete convergence
zbMATH Keywords: discrete pre-Cauchy structure
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/713
OpenAlex ID: W1978857728
description: scientific article; zbMATH DE number 889010