Connections between the completion of normed spaces over non-Archimedean fields and the stability of the Cauchy equation
The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The problem of stability of functional equations was motivated by a question of Ulam being asked in 1940. \textit{M. S. Moslehian} and \textit{T. M. Rassias} [Appl. Anal. Discrete Math. 1, No. 2, 325--334 (2007; Zbl 1257.39019)] proved stability of some functional equations in non-Archimedean spaces. Here the authors provide a close connection between stability results for the Cauchy equation and the completion of a normed space over the rationals endowed with the usual absolute value. In this paper similar results are presented when the valuation of the rationals is a \(p\)-adic valuation.
scientific article; zbMATH DE number 7342913
scientific article