An alternative functional equation on restricted domain

Let X be a real inner product space with dimension greater than 2 and f be a real functional defined on X. In this paper we study an alternative functional equation defined on the whole space and the same equation on the restricted domain of all orthogonal vectors, in order to establish the equivalence between the alternative equation and its conditional form. This research follows previous studies concerning alternative functional equations and the Cauchy functional equation on the restricted domain of the orthogonal vectors.