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Geometry of bilinear forms on a normed space $\mathbb{R}^n$

김성권
대한수학회 (The Korean Mathematical Society)|2023.01|대한수학회지 (Journal of the KMS)|제60권 제1호|pp. 213-225|DOI 10.4134/JKMS.j220277

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참고문헌

키워드

extreme pointsexposed pointsBilinear formssymmetric bilinear forms
주제분류수학

초록

For every $n\geq 2$, let $\mathbb{R}^n_{\|\cdot\|}$ be $\mathbb{R}^n$ with a norm $\|\cdot\|$ 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 ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, where ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ is the space of bilinear forms on $\mathbb{R}^n_{\|\cdot\|}$, and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ is the subspace of ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ consisting of symmetric bilinear forms. Let ${\mathcal F}={\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ or ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$. First we classify the extreme and exposed points of the closed unit ball of ${\mathcal F}$. We also show that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. It is shown that $\qopname\relax o{ext}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}=\qopname\relax o{ext}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ and $\qopname\relax o{exp}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}=\qopname\relax o{exp}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, which expand some results of \cite{18, 23, 28, 29, 35, 38, 40, 41, 43}.

Abstract

For every $n\geq 2$, let $\mathbb{R}^n_{\|\cdot\|}$ be $\mathbb{R}^n$ with a norm $\|\cdot\|$ 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 ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, where ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ is the space of bilinear forms on $\mathbb{R}^n_{\|\cdot\|}$, and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ is the subspace of ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ consisting of symmetric bilinear forms. Let ${\mathcal F}={\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ or ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$. First we classify the extreme and exposed points of the closed unit ball of ${\mathcal F}$. We also show that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. It is shown that $\qopname\relax o{ext}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}=\qopname\relax o{ext}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ and $\qopname\relax o{exp}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}=\qopname\relax o{exp}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, which expand some results of \cite{18, 23, 28, 29, 35, 38, 40, 41, 43}.

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관련 논문AI추천

The unit ball of the space of bilinear forms on $\mathbb{R}^3$ with the supremum normSung Guen Kim대한수학회 · 대한수학회논문집2019 · 제34권 제2호Extremal Problems for ${\mathcal L}_s(^2\mathbb{R}^2_{h(w)})$김성권경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2017 · 제57권 제2호The Geometry of the Space of Symmetric Bilinear Forms on R2 with Octagonal Norm김성권경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2016 · 제56권 제3호THE GEOMETRY OF Ls(3l2∞)김성권대한수학회 · 대한수학회논문집2017 · 제32권 제4호The Geometry of L(2l_{infty}^2)김성근경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2018 · 제58권 제1호The unit ball of L(2R2h(w))김성근대한수학회 · 대한수학회보2017 · 제54권 제2호

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