GEOMETRY OF BILINEAR FORMS ON A NORMED SPACE Rⁿ

For every n ≥ 2, let (formula presented) with a norm (formula presented) such that its unit ball has finitely many extreme points more than 2n. We devote to the description of the sets of extreme and exposed points of the closed unit balls of (formula presented) and (formula presented), where (formula presented) is the (formula presented) space of bilinear forms on (formula presented), and (formula presented) is the subspace of (formula presented) consisting of symmetric bilinear forms. Let (formula presented) or (formula presented) First we classify the extreme and exposed points of the closed unit ball of F. We also show that every extreme point of the closed unit ball of F is exposed. It is shown that ext (formula presented), which expand some results of [18, 23, 28, 29, 35, 38, 40, 41, 43].