Orthogonality and additivity modulo a subgroup
Let \(E\) be a real inner product space of dimension at least 2, \(F\) a topological Abelian group, and \(K\) a discrete subgroup of \(F\). The main result is given in the following theorem: Suppose \(F\) is continuously divisible by 2. If \(f: E\to F\) fulfils the condition \(f(x+y)-f(x)-f(y)\in K\) for all orthogonal \(x,y\in E\) and it is continuous at the origin, then there exist continuous additive functions \(a: \mathbb{R}\to F\) and \(A: E\to F\) such that \(f(x)-a(\| x\|^ 2)- A(x)\in K\) for every \(x\in E\).
scientific article; zbMATH DE number 493230