Functional Equations Related to Inner Product Spaces

and Applied Analysis 3 for all x1, . . . , x2n ∈ V if and only if the odd mapping f : V → W is Cauchy additive, that is, f ( x y ) f x f ( y ) , 2.2 for all x, y ∈ V . Proof. Assume that f : V → W satisfies 2.1 . Letting x1 · · · xn x, xn 1 · · · x2n y in 2.1 , we get nf ( x − x y 2 ) nf ( y − x y 2 ) nf x nf ( y ) − 2nf ( x y 2 ) , 2.3 for all x, y ∈ V . Since f : V → W is odd, 0 nf x nf ( y ) − 2nf ( x y 2 ) , 2.4 for all x, y ∈ V and f 0 0. So 2f ( x y 2 ) f x f ( y ) , 2.5 for all x, y ∈ V . Letting y 0 in 2.5 , we get 2f x/2 f x for all x ∈ V . Thus f ( x y ) 2f ( x y 2 ) f x f ( y ) , 2.6 for all x, y ∈ V . It is easy to prove the converse. For a given mapping f : X → Y , we define Df x1, . . . , x2n : 2n ∑ i 1 f ⎛ ⎝xi − 1 2n 2n ∑