Author:
Sung Guen KimDepartment of Mathematics, Kyungpook National University, Daegu 702-701, South Korea

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Abstract

For n,m≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of ( l n m ) and s ( l n m ) , where ( l n m ) is the space of n-linear forms on m with the supremum norm, and s ( l n m ) is the subspace of ( l n m ) consisting of symmetric n-linear forms. First we classify the extreme points of the unit balls of ( l n m ) and s ( l n m ) , respectively. We show that ext B ( l n m ) ⊂ ext B ( l n m + 1 ) , which answers the question in []. We show that every extreme point of the unit balls of ( l n m ) and s ( l n m ) is exposed, correspondingly. We also show that
ext B s ( l n 2 ) = ext  B ( l n 2 ) s ( l n 2 ) ,
ext  B s ( l 2 m + 1 ) ext  B ( l 2 m + 1 ) s ( l 2 m + 1 ) ,
exp B S ( l n 2 ) = exp B ( l n 2 ) s ( l n 2 )

and exp B s ( l 2 m + 1 ) exp B ( l 2 m + 1 ) s ( l 2 m + 1 ) ,

which answers the questions in [].

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