Functional equations related to inner product spaces (Q1035084)

Summary: Let \(V,W\) be real vector spaces. It is shown that an odd mapping \(f:V\to W\) satisfies \[ \sum^{2n}_{i=1}f(x_i-1/2n\sum_{j=1}^{2n}x_j)=\sum_{i=1}^{2n}f(x_i)-2_nf(1/2n\sum_{i=1}^{2n}x_i) \] for all \(x_1,\dots,x_{2n}\in V\) if and only if the odd mapping \(f:V\to W\) is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.