On some alternative quadratic equations (Q1895253)

From the quadratic functional equation \(f(x + y) + f(x - y) - 2f(x) - 2f(y) = 0\), various alternative equations are derived by grouping in different ways its terms and then equating norms. Some equivalence results are proved in the class of functionals \(f : X \to (\mathbb{R}, |\cdot |)\). Suitable examples concerning operators \(f : X \to (E, |\cdot |)\) with values in normed spaces show that in the more general setting such an equivalence can fail to be true.