Stability and hyperstability of a quadratic functional equation and a characterization of inner product spaces

Abstract

We have proved the Hyers-Ulam stability and the hyperstability of the quadratic functional equation

f(x + y + z) + f(x + y - z) + f(x - y + z) + f(-x + y + z) = 4[f(x) + f(y) + f(z)]

in the class of functions from an abelian group G into a Banach space.