INNER PRODUCT SPACES AND FUNCTIONAL EQUATIONS

Abstract

In [7], Th.M. Rassias introduced the following equality Sigma(n)(i,j=1) parallel to x(i) - x(j)parallel to(2) = 2n Sigma(n)(i=1) parallel to x(i)parallel to(2), Sigma(n)(i=1) x(i)=0 for a fixed integer n >= 3. Let V IV be real vector spaces. In this paper, we show that, if a mapping f : V -> W satisfies Sigma(n)(i,j=1) f(x(i) - x(j)) = 2n Sigma(n)(i=1) f(x(i)) for all x(1), ... , x(n) is an element of V with Sigma(n)(i=1) x(i)=0, then the mapping f: V -> W is realized as the sum of an additive mapping and a quadratic mapping. Furthermore, we prove the generalized Hyers-Ulam stability of the functional equation (0.1) in real Banach spaces.