Inner Product Spaces and Quadratic Functional Equations

In this paper, we prove 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 n parallel to Sigma(n)(i=1)x(i)parallel to(2) + Sigma(n)(i=1)parallel to nx(i) - Sigma(n)(j=1)x(j)parallel to(2) = n(2)Sigma(n)(i=1)parallel to 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 a mapping f : V -> W satisfies nf(Sigma(n)(i=1)x(i)) + Sigma(n)(i=1)f(nx(i) - Sigma(n)(j=1)x(j)) = n(2)Sigma(n)(i=1)f(x(i)), (n > 2) or nf(Sigma(n)(i=1)x(i)) + Sigma(n)(i=1)f(nx(i) - Sigma(n)(j=1)x(j)) = n(2) + n/2 Sigma(n)(i=1)f(x(i)) + n(2) - n/2 Sigma(n)(i=1)f(-x(i)), (n >= 2) for all x(1),..., x(n) is an element of V, then the mapping f : V -> W is Cauchy additive-quadratic. Furthermore, we prove the Hyers-Ulam stability of the above quadratic functional equations in Banach spaces.