On the valuation field invented by A. Robinson and certain structures connected with it
The nonarchimedean valued spherically complete field \(^{\rho}{\mathbb{R}}\) was introduced by \textit{A. Robinson} [Am. Math. Monthly 80, No.6, Part II, 87-109 (1973; Zbl 0269.26020)] as a residue class field of a certain convex subring of the nonstandard real number field \({}^*{\mathbb{R}}\); here \(\rho\) is an arbitrary but fixed positive infinitesimal. The book [\textit{A. H. Lightstone} and \textit{A. Robinson}, Nonarchimedean fields and asymptotic expansions, Amsterdam (1975; Zbl 0303.26013)] is devoted to the field \(^{\rho}{\mathbb{R}}\). The basis of functional analysis over the field \(^{\rho}{\mathbb{R}}\) were outlined by \textit{W. A. J. Luxemburg} [Isr. J. Math. 25, 189-201 (1976; Zbl 0341.12106)], wherein the notion of the \(\rho\)-nonarchimedean hull \(^{\rho}E\) of an internal normed space E was introduced. In the paper under review the notion of the \(\rho\)-nonarchimedean hull is extended to internal Banach-Lie groups. The structure of \(\rho\)- nonarchimedean hulls of normed spaces is somewhat clarified. In particular, it is shown that for any pair E, F of infinite-dimensional standard normed spaces there exists a nonstandard model \({}^*{\mathfrak M}\) of analysis such that the \(\rho\)-nonarchimedean hulls \(^{\rho}E\) and \(^{\rho}F\) taken within this model are isometrically isomorphic.
nonarchimedean valued spherically complete field
infinite-dimensional standard normed spaces
scientific article; zbMATH DE number 4084389
scientific article