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Zeitschrift für Analysis und ihre Anwendungen

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Volume 15, Issue 2, 1996, pp. 529–544
DOI: 10.4171/ZAA/713

Published online: 1996-06-30

Convergence Structures in Numerical Analysis

Siegfried Gähler[1] and D. Matel-Kaminska[2]

(1) Universität Potsdam, Germany
(2) Szczecin University, Poland

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 $$((x_n)_{N'}, x) \in t \Longleftrightarrow (d(q_n(x_n),x))_{N'} \to 0$$ $$ (x_n)_{N’} \in \mathcal C \Longleftrightarrow (q_n (x_n))_{N'} \ \mathrm {is \ a \ Cauchy \ sequence \ in} \ (E,d).$$ 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.

Keywords: Discrete sequences, discrete convergence, discrete (pre-)Cauchy sequences, discrete Cauchy spaces, metric discrete Cauchy spaces, discrete Cauchy spaces of mappings

Gähler Siegfried, Matel-Kaminska D.: Convergence Structures in Numerical Analysis. Z. Anal. Anwend. 15 (1996), 529-544. doi: 10.4171/ZAA/713