ON ORTHOGONALLY ADDITIVE ISOMETRIES

ON ORTHOGONALLY ADDITIVE ISOMETRIES

Let H be a real inner product space. In this paper, we show that if a mapping f : H → H satisfies f(x + y) = f(x) + f(y) for all x, y ∈ H with x ⊥ y and ∥f(x)∥ = ∥x∥ for all x,∈ H, then f is an additive mapping.