Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GEOMETRY OF BILINEAR FORMS ON A NORMED SPACE ℝn

  • Sung Guen Kim (Department of Mathematics Kyungpook National University)
  • Received : 2022.06.02
  • Accepted : 2022.09.26
  • Published : 2023.01.01
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Abstract

For every n ≥ 2, let ℝn‖·‖ be Rn with a norm ‖·‖ 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 𝓛(2ℝn‖·‖) and 𝓛𝒮(2ℝn‖·‖), where 𝓛(2ℝn‖·‖) is the space of bilinear forms on ℝn‖·‖, and 𝓛𝒮(2ℝn‖·‖) is the subspace of 𝓛(2ℝn‖·‖) consisting of symmetric bilinear forms. Let 𝓕 = 𝓛(2ℝn‖·‖) or 𝓛𝒮(2ℝn‖·‖). 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(2ℝn‖·‖) = ext B𝓛(2ℝn‖·‖) ∩ 𝓛𝒮(2ℝn‖·‖) and exp B𝓛𝒮(2ℝn‖·‖) = exp B𝓛(2ℝn‖·‖) ∩ 𝓛𝒮(2ℝn‖·‖), which expand some results of [18, 23, 28, 29, 35, 38, 40, 41, 43].

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References

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