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 n ≥ 2

\sum_{i=1}^n \| x_i - \frac{1}{n} \sum_{j=1}^n x_j \|^2

= \sum_{i=1}^n \| x_i \|^2

- n \| \frac{1}{n} \sum_{i=1}^n x_i \|^2

holds for all x_1, ···, x_n ∈V.

Let V,W be real vector spaces. It is shown that if an even mapping f : V → W satisfies

\sum_{i=1}^{2n} f (x_i - \frac{1}{2n} \sum_{j=1}^{2n} x_j )

= \sum_{i=1}^{2n} f(x_i) - 2n f (\frac{1}{2n} \sum_{i=1}^{2n} x_i )

for all x_1, ···, x_{2n} ∈ V, then the even mapping f : V → W is quadratic.

Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.