Functional equations associated with inner product spaces

Abstract

In \cite{ra84}, Th.M. Rassias proved that the norm defined over a real vector space $V$ is induced by an inner product if and only if for a fixed integer $n \ge 2$ \begin{eqnarray*} n \left\|\frac{1}{n}\sum_{i=1}^n x_i\right\|^2 + \sum_{i=1}^n \left\|x_i - \frac{1}{n} \sum_{j=1}^n x_j\right\|^2 = \sum_{i=1}^n\|x_i\|^2 \end{eqnarray*} holds for all $x_1, \cdots, x_n \in V$.

Let $V, W$ be real vector spaces. It is shown that if a mapping $f : V \rightarrow W$ satisfies \begin{eqnarray} n f\left(\frac{1}{n}\sum_{i=1}^{n} x_i\right) + \sum_{i=1}^{n} f\left(x_i - \frac{1}{n} \sum_{j=1}^{n} x_j\right) = \sum_{i=1}^{n}f(x_i) \end{eqnarray} for all $x_1, \cdots, x_{n} \in V$, then the mapping $f : V \rightarrow W$ satisfies \begin{eqnarray} 2 f\left(\frac{x+y}{2}\right) + f\left( \frac{x-y}{2}\right) + f\left(\frac{y-x}{2}\right) = f(x) + f(y) \end{eqnarray} for all $x, y \in V$.

Furthermore, we prove the generalized Hyers-Ulam stability of the functional equation {\rm (0.2)} in real Banach spaces.