Inner product spaces and quadratic functional equations

Abstract

In this paper, we prove that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer n ≥ 2 (Formula presented) holds for all x1, …,xn ∈ V. Let V, W be real vector spaces. It is shown that if a mapping f: V → W satisfies (Formula presented) or (Formula presented) for all x1, …, xn ∈ V, then the mapping f: V → W is Cauchy additive-quadratic. Furthermore, we prove the Hyers-Ulam stability of the above quadratic functional equations in Banach spaces.