• Kyungpook Mathematical Journal
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• 제57권2호
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• pp.223-232
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• 2017

We classify the extreme and exposed symmetric bilinear forms of the unit ball of the space of symmetric bilinear forms on ${\mathbb{R}}^2$ with hexagonal norms. We also show that every extreme symmetric bilinear forms of the unit ball of the space of symmetric bilinear forms on ${\mathbb{R}}^2$ with hexagonal norms is exposed.

• Kyungpook Mathematical Journal
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• 제54권3호
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• pp.341-347
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• 2014

We classify the exposed symmetric bilinear forms of the unit ball of $\mathcal{L}_s(^2d_*(1,{\omega})^2)$.

• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.781-791
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• 2016

Let $d_*(1,w)^2 ={\mathbb{R}}^2$ with the octagonal norm of weight w. It is the two dimensional real predual of Lorentz sequence space. In this paper we classify the smooth points of the unit ball of the space of symmetric bilinear forms on $d_*(1,w)^2$. We also show that the unit sphere of the space of symmetric bilinear forms on $d_*(1,w)^2$ is the disjoint union of the sets of smooth points, extreme points and the set A as follows: $$S_{{\mathcal{L}}_s(^2d_*(1,w)^2)}=smB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}extB_{{\mathcal{L}}_s(^2d_*(1,w)^2)}{\bigcup}A$$, where the set A consists of $ax_1x_2+by_1y_2+c(x_1y_2+x_2y_1)$ with (a = b = 0, $c={\pm}{\frac{1}{1+w^2}}$), ($a{\neq}b$, $ab{\geq}0$, c = 0), (a = b, 0 < ac, 0 < ${\mid}c{\mid}$ < ${\mid}a{\mid}$), ($a{\neq}{\mid}c{\mid}$, a = -b, 0 < ac, 0 < ${\mid}c{\mid}$), ($a={\frac{1-w}{1+w}}$, b = 0, $c={\frac{1}{1+w}}$), ($a={\frac{1+w+w(w^2-3)c}{1+w^2}}$, $b={\frac{w-1+(1-3w^2)c}{w(1+w^2)}}$, ${\frac{1}{2+2w}}$ < c < ${\frac{1}{(1+w)^2(1-w)}}$, $c{\neq}{\frac{1}{1+2w-w^2}}$), ($a={\frac{1+w(1+w)c}{1+w}}$, $b={\frac{-1+(1+w)c}{w(1+w)}}$, 0 < c < $\frac{1}{2+2w}$) or ($a={\frac{1=w(1+w)c}{1+w}}$, $b={\frac{1-(1+w)c}{1+w}}$, $\frac{1}{1+w}$ < c < $\frac{1}{(1+w)^2(1-w)}$).

• 대한수학회지
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• 제60권1호
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• pp.213-225
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• 2023

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 𝓛_𝒮(²ℝⁿ_‖·‖), where 𝓛(²ℝⁿ_‖·‖) is the space of bilinear forms on ℝⁿ_‖·‖, and 𝓛_𝒮(²ℝⁿ_‖·‖) is the subspace of 𝓛(²ℝⁿ_‖·‖) consisting of symmetric bilinear forms. Let 𝓕 = 𝓛(²ℝⁿ_‖·‖) or 𝓛_𝒮(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 B_{𝓛𝒮(²ℝⁿ‖·‖)} = ext B_{𝓛(²ℝⁿ‖·‖)} ∩ 𝓛_𝒮(²ℝⁿ_‖·‖) and exp B_{𝓛𝒮(²ℝⁿ‖·‖)} = exp B_{𝓛(²ℝⁿ‖·‖)} ∩ 𝓛_𝒮(²ℝⁿ_‖·‖), which expand some results of [18, 23, 28, 29, 35, 38, 40, 41, 43].

• Kyungpook Mathematical Journal
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• 제53권2호
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• pp.295-306
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• 2013

First we present the explicit formula for the norm of a symmetric bilinear form on the 2-dimensional real predual of the Lorentz sequence space $d_*(1,w)^2$. Using this formula, we classify the extreme points of the unit ball of $\mathcal{L}_s(^2d_*(1,w)^2)$.

• 대한수학회지
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• 제48권1호
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• pp.1-12
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• 2011

The purpose of this paper is to study pseudo-Riemannian algebras, which are algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. We nd that pseudo-Riemannian algebras whose left centers are isotropic play a curial role and show that the decomposition of pseudo-Riemannian algebras whose left centers are isotropic into indecomposable non-degenerate ideals is unique up to a special automorphism. Furthermore, if the left center equals the center, the orthogonal decomposition of any pseudo-Riemannian algebra into indecomposable non-degenerate ideals is unique up to an isometry.

• Kyungpook Mathematical Journal
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• 제60권3호
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• pp.485-505
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• 2020

We present a complete description of all the extreme points of the unit ball of 𝓛_s(²𝑙³_∞) which leads to a complete formula for ║f║ for every f ∈ 𝓛_s(²𝑙³_∞)^∗. We also show that $extB_{{\mathcal{L}}_s(^2l^3_{\infty})}{\subset}extB_{{\mathcal{L}}_s(^2l^n_{\infty})}$ for every n ≥ 4. Using the formula for ║f║ for every f ∈ 𝓛_s(²𝑙³_∞)^∗, we show that every extreme point of the unit ball of 𝓛_s(²𝑙³_∞) is exposed. We also characterize all the smooth points of the unit ball of 𝓛_s(²𝑙³_∞).