# Convergence Structures in Numerical Analysis

### Siegfried Gähler

Universität Potsdam, Germany### D. Matel-Kaminska

Szczecin University, Poland

## Abstract

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 $E_1, E_2,...$ 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 $\mathbb N = {1,2,…}$. 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 $(((E_1, E_2,…), E); (t, \mathcal C))$. 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

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 sufficient condition for its completeness are given.

## Cite this article

Siegfried Gähler, D. Matel-Kaminska, Convergence Structures in Numerical Analysis. Z. Anal. Anwend. 15 (1996), no. 2, pp. 529–544

DOI 10.4171/ZAA/713