Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 57: Issue 3
The unit balls of ℒ(nl∞m) and ℒs(nl∞m)
Author:
Sung Guen Kim Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea

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Pages:
267–283
Online Publication Date:
20 Oct 2020
Publication Date:
20 Oct 2020
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2020.57.3.1470
Keywords:
Primary 46A22; n-linear forms; symmetric n-linear forms; extreme points; exposed points
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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(ln∞m) and ℒs(ln∞m), where ℒ(ln∞m) is the space of n-linear forms on ℝm with the supremum norm, and ℒs(ln∞m) is the subspace of ℒ(ln∞m) consisting of symmetric n-linear forms. First we classify the extreme points of the unit balls of ℒ(ln∞m) and ℒs(ln∞m), respectively. We show that ext Bℒ(ln∞m) ⊂ ext Bℒ(ln∞m+1), which answers the question in [32]. We show that every extreme point of the unit balls of ℒ(ln∞m) and ℒs(ln∞m) is exposed, correspondingly. We also show that
extBℒs(ln∞2)=ext Bℒ(ln∞2)∩ℒs(ln∞2),
ext Bℒs(l2∞m+1)≠ext Bℒ(l2∞m+1)∩ℒs(l2∞m+1),
expBℒS(ln∞2)=expBℒ(ln∞2)∩ℒs(ln∞2)

and expBℒs(l2∞m+1)≠expBℒ(l2∞m+1)∩ℒs(l2∞m+1),

which answers the questions in [31].

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Studia Scientiarum Mathematicarum Hungarica
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Studia Scientiarum Mathematicarum Hungarica
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