Let P1, …, Pn be polynomials in one or several real or complex variables. Several authors, working with a variety of norms, have given estimates for a constant M depending only on the degrees of P1, …, Pn such that

∥P1∥…∥Pn∥ [les ]M∥P1 … Pn∥.

In this paper we show that inequalities of this type are valid for polynomials on any complex Banach space. Our method provides optimal constants.

We also derive analogous inequalities for polynomials on real Banach spaces, but the constants we obtain are generally not optimal. The search for optimal constants does however lead to an interesting open problem in Hilbert space geometry.

When we restrict attention to products of linear functionals, we find new characterizations of complex [lscr ]n1 (n[ges ]2), real [lscr ]21, and real Hilbert space.