A note on the Lipschitz condition in metric spaces

Abstract

where the summation runs over all ni, * *, nGS, and k1, * * , kq ESq, SQ being the set of all finite sequences of positive integers with q elements; juxtaposition of the f's of course means composition. Then d * satisfies all of the axioms for a metric on M.. The proof will be complete when we show that (1) d * is equivalent to d. on M. and (2) fjk: M1-*Mj is Lipschitzian relative to d* and dj*. To show (1) let yCM. and {yn} be an infinite sequence in M3. Suppose limn d*(y, yn) =0. Then limn d8(y, yn) =0 since d,(y, yn) <d *(y, yn). Conversely suppose limn d.(y, yn) = 0-. We note first that