GEOMETRY OF BILINEAR FORMS ON A NORMED SPACE Rn

초록

For every n > 2, let Rnk center dot k be Rn with a norm center dot 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 G(2Rnk center dot k) and G3(2Rnk center dot k), where G(2Rnk center dot k) is the space of bilinear forms on Rnk center dot k, and G3(2Rnk center dot k) is the subspace of G(2Rnk center dot k) consisting of symmetric bilinear forms. Let F = G(2Rnk center dot k) or G3(2Rn k center dot k). 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 BLs(2RnIl center dot Il) = ext BL(2RnIl center dot Il) nG3(2Rnk center dot k) and exp BLs(2RnIl center dot Il) = exp BL(2RnIl center dot Il) n G3(2Rnk center dot k), which expand some results of [18,23, 28, 29, 35, 38, 40, 41, 43].

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