Metric definition of the linear structure

Abstract

Mazur and Ulam [1 ] proved that the metric of a real normed linear space determines the linear operations. The following converse of an elementary theorem gives an explicit characterization of convex combinations in terms of the metric d of such a space. THEOREM. If 0 < a < 1 and if x, y, and z are members of a real normed linear space such that for each w in the space d(w, z) ? ad(w, x) + (1 -a)d(w, y) then z =ax+ (I1-a)y. PROOF. We may assume z = 0 and a < 1/2. Taking a = 1/2, suppose 2d(w, 0) !d(w, x) +d(w, y) for all w. Then 0 is in the set T of those nonnegative integers t for which, for each positive integer m, (2m + t)d(x + y, 0) ? d(mx + my, x) + d(mx + my, y). Choosing t in T, observe that if m is a positive integer then so is 2m, and (4m + t)d(x + y, 0) ! d(2mx + 2my, x) + d(2mx + 2my, y). Setting w = (2m 1)x+2my in the hypothesis leads to d(2mx + 2my, x) g d(mx + my, x) + (m 1/2)d(x + y, 0). This last statement also holds with x and y interchanged, and from these inequalities we conclude that (4m + t)d(x + y, 0) g d(mx + my, x) + d(mx + my, y) + (2ml)d(x + y, 0). Therefore (2m+t+l)d(x+y, O)<d(mx+my, x)+d(mx+my, y). By induction, T contains all nonnegative integers. In particular, if t is a nonnegative integer then (2 + t)d(x + y, O) g d(y, O) + d(x, 0). This implies x +y = 0, as desired. Now if 0<a<1/2 and if