Conditional functional equations on particular normed spaces

Let X be a real normed space with dimension greater than 2 and f be a real functional defined on X. Applying some ideas from the studies made on the conditional Cauchy functional equation on the restricted domain of the vectors of equal norm, the isosceles orthogonal vectors and the pythagorean orthogonal vectors, the conditional quadratic equation and the D’Alembert one have been studied in this paper, in order to describe their solutions. Particular normed spaces are introduced for this aim.