Extreme polynomials and multilinear forms on l(1)

Given a 2-homogeneous polynomial P(x, y) = ax(2) + by(2) + cxy with real co efficients, 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), resp ectively. 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 extr eme 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 fi nd 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 extre me point. We also characterize extreme points and strongly extreme points o f the unit ball of L((m)l(1)) or L(L-m(1)[0, 1]). (C) 1998 Academic Press.