Pointwise convergence of Cauchy sequences
If G is a normed space whose elements are real-valued functions on a set X it is sometimes interesting or even important to know whether the completion of G can also be realized as a space of real-valued functions on X. \textit{J. W. Brace} and \textit{J. D. Thomison} used in Ill. J. Math. 22, 581-602 (1978; Zbl 0384.46016) and in Ill. J. Math. 23, 93-100 (1979; Zbl 0391.46022) the notion of \({\mathcal F}\)-topology to study the problem: If \({\mathcal F}\) is a filter on X, then (G,\({\mathcal F})\) is the topological group whose zero-neighbourhood basis is given by \[ \{\{f\in G:\quad there\quad is\quad an\quad F\in {\mathcal F}\quad with\quad f(F)\subseteq [- \epsilon,\epsilon]\},\quad \epsilon >0\}. \] They gave a representation of the desired form for \(G=(G,{\mathcal F})\) if X is linear, \({\mathcal F}\) convex and circled and moreover every convergent sequence in G converges uniformly on some \(F\in {\mathcal F}\). If the condition is relaxed, then additional conditions of another kind are required. In this paper the autor considers Cauchy-sequences instead of convergent sequences and gets similar results.
pointwise Cauchy filter convergence space
subuniform Cauchy filter convergence space
subpointwise Cauchy filter convergence space
scientific article; zbMATH DE number 4071660