On n-collinear elements and Riesz theorem

In this paper, we prove that the n-collinear elements x 1 ; x 2 ; : . . . ; x n ; u satisfy some special relations in an n-normed space X. Further, we prove that u = x 1 + . . . + x n n is the only unique element in the n-normed space X such that x 1 ; x 2 ; . . . ; x n ; u are n-collinear elements in X satisfying some specified inequalities. Moreover, we prove that the Riesz theorem holds when X is a linear n-normed space.