JournalszaaVol. 15, No. 2pp. 529–544

Convergence Structures in Numerical Analysis

  • Siegfried Gähler

    Universität Potsdam, Germany
  • D. Matel-Kaminska

    Szczecin University, Poland
Convergence Structures in Numerical Analysis cover
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Abstract

The paper deals — under the viewpoint of topology — with discrete Cauchy spaces, which are spaces where a discrete Cauchy structure (t,C)(t,\mathcal C) (with tt being a discrete convergence and C\mathcal C being a discrete pre-Cauchy structure) is defined. More precisely, let E1,E2,...E_1, E_2,... and EE be arbitrary sets and let S\mathcal S denote the set of all discrete sequences (xn,nN(x_n,_{n \in N'} with xn,En(nN)x_n, \in E_n (n \in N’) and with NN’ being an infinite subset of N=1,2,\mathbb N = {1,2,…}. Then tt and C\mathcal C are certain subsets of (S,E)(\mathcal S, E) respectively of S\mathcal S, which in a certain sense are assumed to be compatible. The paper gives properties of tt and C\mathcal C and among others is devoted to the problem of completion of discrete Cauchy spaces (((E1,E2,),E);(t,C))(((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 EE is equipped with a metric dd and that there exist mappings qn:EnE(nN)q_n : E_n \to E (n \in \mathbb N) — the discrete Cauchy structure (t,C)(t, \mathcal C) is defined by

((xn)N,x)t(d(qn(xn),x))N0((x_n)_{N'}, x) \in t \Longleftrightarrow (d(q_n(x_n),x))_{N'} \to 0
(xn)NC(qn(xn))N is a Cauchy sequence in (E,d).(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)(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