Functional Equations Related to Inner Product Spaces

Abstract

Let $V,W$ be real vector spaces. It is shown that an odd mapping $f:V\to W$ satisfies ${\displaystyle {\sum }_{i-1}^{2n}f\left(x_i-1/2n{\displaystyle {\sum }_{j=1}^{2n}{x}_j}\right)=}{\displaystyle {\sum }_{i=1}^{2n}f\left(x_i\right)-2nf\left(1/2n{\displaystyle {\sum }_{i=1}^{2n}{x}_i}\right)}$ 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.