Functional Equations Related to Inner Product Spaces

Abstract

Let V,W be real vector spaces. It is shown that an odd mapping f : V -> W satisfies Sigma(2n)(i-1)f(x(i) - 1/wn Sigma(2n)(j=1)x(j)) = Sigma(2n)(i=1)f(x(i)) - 2nf(1/2n Sigma(2n)(i=1)x(i)) for all x(1), ..., x(2n) is an element of V if and only if the odd mapping f : V -> W is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.