Extreme polynomials and multilinear forms on l(1)

Abstract

Given a 2-homogeneous polynomial P(x, y) = ax(2) + by(2) + cxy with real coefficients, let parallel to P parallel to(r) and parallel to P parallel to(c) denote the norms of P on the real and complex Banach space l(1)(2), respectively. We show parallel to P parallel to(r) = parallel to P parallel to(c), and obtain a sufficient and necessary condition on the coefficients a, b, and c for P to have norm 1. Applying these results, we characterize extreme points of the unit ball of P((2)l(1)) for the real Banach space l(1)(2) and examine them for the complex Banach space l(1)(2). We apply them to find extreme points and strongly extreme points of the unit ball of P((2)l(1)) and get an extremal 2-homogeneous polynomial on l(1) that is not an extreme point. We also characterize extreme points and strongly extreme points of the unit ball of L((m)l(1)) or L(L-m(1)[0, 1]). (C) 1998 Academic Press.