On some characterization of the absolute value of an additive function - Tabor, Józef (Annales Mathematicae Silesianae, 1994)
Let G be an abelian group, let K be the real or complex field, let X be a normed space over ????, and let u∈X such that ‖u‖=1 be given. We assume that there exists a subspace X1 of X such that X = Lin(u) ⊕ X1 and ‖αu+x1‖ ≥ max(|α|, ‖x1‖) for α∈K , x1∈X1. Then we prove that the general solution f: G→K of the equation f(x+y) + f(x-y) + ‖f(x+y)-f(x-y)‖u = 2f(x) + 2f(y) for x,y∈G is given by the formula f(x) = |a(x)|u for x∈G, where a: G→ℝ is an additive function.