Functional Equations Related to Inner Product Spaces

Let V,W be real vector spaces. It is shown that an odd mappingf:V→W satisfies ∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi) for all x1,…,x2n∈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 equationin real Banach spaces