Geometry of multilinear forms on a normed space Rm
DOI:
https://doi.org/10.3842/umzh.v76i5.7476Keywords:
multilinear forms, extreme points, exposed points.Abstract
UDC 514.1
For every m≥2, let Rm‖⋅‖ be Rm with a norm ‖⋅‖ such that its unit ball has finitely many extreme points. For every n≥2, we focus our attention on the description of the sets of extreme and exposed points of the closed unit balls of L(nRm‖⋅‖) and Ls(nRm‖⋅‖), where L(nRm‖⋅‖) is the space of n-linear forms on Rm‖⋅‖ and Ls(nRm‖⋅‖) is the subspace of L(nRm‖⋅‖) formed by symmetric n-linear forms. Let F=L(nRm‖⋅‖) or Ls(nRm‖⋅‖). First, we show that the number of extreme points of the unit ball of Rm‖⋅‖ is greater than 2m. By using this fact, we classify the extreme and exposed points of the closed unit ball of F, respectively. It is shown that every extreme point of the closed unit ball of F is exposed. We obtain the results of [Studia Sci. Math. Hungar., 57, No. 3, 267–283 (2020)] and extend the results of [Acta Sci. Math. Szedged, 87, No. 1-2, 233–245 (2021) and J. Korean Math., Soc., 60, No. 1-2, 213–225 (2023)].
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