# 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 $(t,C)$ (with $t$ being a discrete convergence and $C$ being a discrete pre-Cauchy structure) is defined. More precisely, let $E_{1},E_{2},...$ and $E$ be arbitrary sets and let $S$ denote the set of all discrete sequences $(x_{n},_{n∈N_{′}}$ with $x_{n},∈E_{n}(n∈N’)$ and with $N’$ being an infinite subset of $N=1,2,…$. Then $t$ and $C$ are certain subsets of $(S,E)$ respectively of $S$, which in a certain sense are assumed to be compatible. The paper gives properties of $t$ and $C$ and among others is devoted to the problem of completion of discrete Cauchy spaces $(((E_{1},E_{2},…),E);(t,C))$. 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 $q_{n}:E_{n}→E(n∈N)$ — the discrete Cauchy structure $(t,C)$ 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