The failure of Bernstein's theorem for polynomials on C(K) spaces (Q1101625)

Bernstein's theorem states that if \(p: {\mathbb{C}}\to {\mathbb{C}}\) is a polynomial of degree m, then its derivative p' satisfies \(\| p'\|_{\infty}\leq m\| p\|_{\infty}\), where \(\| \|_{\infty}\) denotes the supremum norm taken over the unit disc. \textit{L. A. Harris}, Bounds on the derivatives of holomorphic functions of vectors, in C. R. Colloq. d'analyse, Rio de Janeiro, 1972, 145-163 (1975; Zbl 0315.46040) proved an analogous inequality for the Fréchet derivative of a polynomial on Hilbert space. He also asked whether there is a similar result for polynomials on C(K) spaces. The purpose of the present note is to give a negative answer, even for polynomials of degree two.