Convergence structures in numerical analysis
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 sufficient condition for its completeness are given.
scientific article; zbMATH DE number 889010
scientific article