Let T be a non expansive mapping on a normed linear space X. We show that there exists a linear functional f, with ||f|| = 1, such that, for all x in X, the Iimit, as n goes to infinity, of  f(T"x/n) equals the limit of IIT"x/nll=a, where a=inf_{y}IITy-yli. This means, if X is reflexive, that there is a face F of the ball of radius a to which T"x/n converges weakly to F for all x  if X is strictly convex as well as reflexive, the convergence is to a point; and if X satisfies the stronger condition that its dual has Frechet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansive T.