Extreme bilinear forms on Rⁿ with the supremum norm

For every n≥ 2 this paper is devoted to the description of the sets of extreme points of the closed unit balls of L(2l∞n) and Ls(2l∞n), where L(2l∞n) is the space of bilinear forms on Rⁿ with the supremum norm, and Ls(2l∞n) is the subspace of L(2l∞n) consisting of symmetric bilinear forms. First we obtain an elegant formula for calculating the norm of a given bilinear form T∈L(2l∞n). We present a characterization of the sets extBL(2l∞n) and extBLs(2l∞n), correspondingly. We obtain a sufficient condition for a given bilinear form T∈extBLs(2l∞n) to be considered as an element of extBLs(2l∞n+1). As applications we show that for every n≥ 3 the relations extBL(2l∞2)⊂extBL(2l∞n) and extBLs(2l∞2)⊂extBLs(2l∞n) hold true. In addition it is shown that for n≥ 3 , extBLs(2l∞n)⊄extBL(2l∞n) in contrast to the case n= 2.