Given a 2-homogeneous polynomialP(x,y)=ax2+by2+cxywith real coefficients, let Prand Pcdenote the norms ofPon the real and complex Banach spacel21, respectively. We show Pr=Pc, and obtain a sufficient and necessary condition on the coefficientsa,b, andcforPto have norm 1. Applying these results, we characterize extreme points of the unit ball of P(2l1) for the real Banach spacel21and examine them for the complex Banach spacel21. We apply them to find extreme points and strongly extreme points of the unit ball of P(2l1) and get an extremal 2-homogeneous polynomial onl1that is not an extreme point. We also characterize extreme points and strongly extreme points of the unit ball of L(ml1) or L(mL1[0,1]).
Bibliographical noteFunding Information:
*Supported by KOSEF grant 961-0102-014-2 and the Basic Science Research Institute Program, Ministry of Education, BSRI-N96089. ²Research done at the Basic Science Research Institute, POSTECH.
All Science Journal Classification (ASJC) codes
- Applied Mathematics