A Bernstein]Markov Theorem for Normed Spaces

Let X and Y be real normed linear spaces and let f: XªR be a non-negative function satisfying f xqy.Ff x.q5y5 for all x, ygX. We show that there exist optimal constants cm, k such that if P: XªY is any polynomial satisfying 5P x.5Ff x.m for all xgX, then 5DˆkP x.5Fcm, kf x.myk whenever xgX and 0FkFm. We obtain estimates for these constants and present applications to polynomials and multilinear mappings in normed spaces