Extreme and exposed points of ℒ(^n ɭ^2_∞ ) and ℒs (^n ɭ^2_∞)

Abstract

For every n ≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of ℒ(n ɭ2∞ ) and ℒs(n ɭ2∞ ), where ℒ(n ɭ2∞ ) is the space of n-linear forms on ℝ2 with the supremum norm, and ℒs(n ɭ2∞ ) is the subspace of ℒ(n ɭ2∞ ) consisting of symmetric n-linear forms. First we classify the extreme points of the closed unit balls of ℒ(n ɭ2∞ ) and ℒs(n ɭ2∞ ) correspondingly. As corollaries we obtain |ext Bℒ(n ɭ2∞ ) | = 2(2n) and =|ext Bℒs(n ɭ2∞ ) | =2n+1. We also show that exp BL(n ɭ2∞ ) =ext Bℒ(n ɭ2∞ ) and exp Bℒs(n ɭ2∞ ) =ext Bℒs(n ɭ2∞ ) .