Kaiser, Z. On stability of the Cauchy equation in normed spaces over fields with valuation. (English) Zbl 1084.39026 Publ. Math. Debr. 64, No. 1-2, 189-200 (2004). The author generalizes some known results on the stability of the Cauchy equation. His main statement is the following.Let \((X,\| \,\,\| _{1})\) be a normed space over a field \(F\) of characteristic zero with a valuation \(| \,\,| _{F},\, (Y,\| \,\, \| _{2})\) be a Banach space over a field \(K\) of characteristic zero with a valuation \(| \,\,| _{K},\, f:X\to Y\) and \(\alpha\) be a real number. If the function \(f\) satisfies \[ \| f(x+y)-f(x)-f(y)\| _{2}\leq L\max \{\| x\| ^{\alpha}_{1}, \| y\| ^{\alpha}_{1}\} \] for all \(x,y \in X\) and for some non-negative real number \(L\) and there exists a positive integer \(s\) such that \(| s| ^{\alpha}_{F}\neq | s| _{K},\) then there exist a unique additive function \(g:X\to Y\) and a real number \(C\) for which \[ \| f(x)-g(x)\| _{2}\leq C\| x\| ^{\alpha}_{1} \] for all \(x\in X.\) Here \(0^{\alpha}=0\) for \(\alpha\neq 0\) and \(0^{0}=1\). Reviewer: Gyula Maksa (Debrecen) Cited in 4 Documents MSC: 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 12J10 Valued fields 12J12 Formally \(p\)-adic fields Keywords:Cauchy’s functional equation; stability; \(p\)-adic fields; fields with valuation; Banach space PDFBibTeX XMLCite \textit{Z. Kaiser}, Publ. Math. Debr. 64, No. 1--2, 189--200 (2004; Zbl 1084.39026)