Geometry of Multilinear Forms on a Normed Space Rm

초록

For every m = 2, let R mk center dot k be Rm with a norm k center dot k 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 n R mk center dot k and Ls(nR mk center dot k), where L nR mk center dot k is the space of n-linear forms on R mk center dot k and Ls(n R mk center dot k) is the subspace of L n R mk center dot k formed by symmetric n-linear forms. Let F = L n R mk center dot k or Ls nR mk center dot k. First, we show that the number of extreme points of the unit ball in R mk center dot k is greater than 2m. By using this fact, we classify the extreme and exposed points of the closed unit ball in F, respectively. It is shown that every extreme point of the closed unit ball in F is exposed. We obtain the results of [Studia Sci. Math. Hungar., 57, No. 3, 267 (2020)] and extend the results from [Acta Sci. Math. (Szeged), 87, Nos. 1-2, 233 (2021) and J. Korean Math. Soc., 60, No. 1-2, 213 (2023)].

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