In [7], 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 $n{geq}2$ $${sum_{i=1}^{n}}left|x_i-{frac{1}{n}}{sum_{j=1}^{n}}x_j ight|^2={sum_{i=1}^{n}}{parallel}x_i{parallel}^2-nleft|{frac{1}{n}}{sum_{i=1}^{n}}x_i ight|^2$$ holds for all $x_1$, ${cdots}$, $x_n{in}V$. Let V, W be real vector spaces. It is shown that if an even mapping $f:V{ ightarrow}W$ satisfies $$(0.1);{sum_{i=1}^{2n}f}(x_i-{frac{1}{2n}}{sum_{j=1}^{2n}}x_j)={sum_{i=1}^{2n}}f(x_i)-2nf({frac{1}{2n}}{sum_{i=1}^{2n}}x_i)$$ for all $x_1$, ${cdots}$, $x_{2n}{in}V$, then the even mapping $f:V{ ightarrow}W$ is quadratic. Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.