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
depending only on the degrees of
P1, …, Pn such
[les ]M∥P1 … Pn∥.
In this paper we show that inequalities of this type are valid for polynomials
complex Banach space. Our method provides optimal constants.
We also derive analogous inequalities for polynomials on real Banach
the constants we obtain are generally not optimal. The search for optimal
does however lead to an interesting open problem in Hilbert space geometry.
When we restrict attention to products of linear functionals, we find
characterizations of complex [lscr ]n1 (n[ges ]2),
real [lscr ]21, and real Hilbert space.