QUADRATIC MAPPINGS ASSOCIATED WITH INNER PRODUCT SPACES

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 Sigma(n)(i=1) parallel to x(i) - 1/n Sigma(n)(j=1) x(j)parallel to(2) = Sigma(n)(i=1) parallel to x(i)parallel to(2) - n parallel to 1/n Sigma(n)(i=1) x(i)parallel to(2) holds for all x(1), . . . , x(n) is an element of V. Let V, W be real vector spaces. It is shown that if an even mapping f : V -> W satisfies (0.1) Sigma(2n)(i=1) f (x(i) - 1/2n Sigma(2n)(j=1) x(j)) = Sigma(2n)(i=1) f(x(i)) - 2nf (1/2n Sigma(2n)(i=1) x(i)) for all x(1) , . . . , x(2n) is an element of 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.