The failure of Bernstein's theorem for polynomials on C( K) spaces

Abstract

Bernstein's theorem asserts that if p: C → C is a polynomial of degree m, then its derivative p′ satisfies the inequality ∥ p′∥, ⩽ m ∥ p∥, where the symbol | | ∞ denotes the supremum norm taken over the unit disc. Harris [2] proved an analogous inequality for the Fréchet derivative of polynomials on Hilbert space. In his commentary to problem 73 in the Scottish Book ( R. D. Mauldin, Ed., Birkhäuser, Boston, pp. 144–145, 1981), he asked whether there is a similar result for polynomials on C( K) spaces. The purpose of this note is to give a negative answer, even for polynomials of degree 2.