On the Cauchy difference on normed spaces
Let \(f\) be a function from a real linear normed space \(X\) into a topological group \((G,+)\). Fixed a discrete normal subgroup \(K\) of \(G\), one may consider the condition \(f(x+y)- f(x)- f(y) \in K\) whenever \(|x|= |y|\). The author of this paper studies this condition and finds several theorems showing that under some regularity assumptions on \(f\) (continuity at a point, measurability in a precise sense, \(\dots)\) and some natural assumptions on \(G,K\) and \(X\), there exists a continuous additive function \(A\) from \(X\) into \(G\) such that \(f(x)-A(x)\in K\).
scientific article; zbMATH DE number 977862