A Bernstein–Markov Theorem for Normed Spaces

Abstract

LetXandYbe real normed linear spaces and let φ:X→R be a non-negative function satisfying φ(x+y)≤φ(x)+‖y‖ for allx,y∈X. We show that there exist optimal constantscm,ksuch that ifP:X→Yis any polynomial satisfying ‖P(x)‖≤φ(x)mfor allx∈X, then ‖D̂kP(x)‖≤cm,kφ(x)m−kwheneverx∈Xand 0≤k≤m. We obtain estimates for these constants and present applications to polynomials and multilinear mappings in normed spaces.