We would like to sincerely thank Professor Saburou Saitoh for his encouragement.

ABSTRACT. Replacing the triangle inequality, in the definition of a norm, by ‖x+ y‖q ≤ 2q−1 (‖x‖q + ‖y‖q), we introduce the notion of a q-norm. We establish that every q-norm is a norm in the usual sense, and that the converse is true as well