On some conditional functional equations - Szostok, Tomasz (Annales Mathematicae Silesianae, 2003)
Let X, Y be real linear spaces. We are looking for a function G : X2→ℝ such that the equation f(x+y) = G(x,y)[f(x)+f(y)] is equivalent to the orthogonal Cauchy equation x⟂y ⇒ f(x+y)=f(x)+f(y). Several kinds of orthogonalities are considered. The quotient ‖(x-y)/(x+y)‖ closely connected with the James orthogonality plays here a distinguished role. Similar problems are considered for the Ptolemaic equation x⟂y ⇒ f(x+y)f(x-y) = f(x)2+f(y)2. As a result a characterization of inner product spaces is obtained.