QUADRATIC MAPPINGS ASSOCIATED WITH INNER PRODUCT SPACES

Abstract

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