Abstract

An element (x1,…,xn)∈En(x_1, \ldots, x_n)\in E^n is called norming point of T∈L(nE)T\in {\mathcal L}(^n E) if
∥x1∥=⋯=∥xn∥=1\|x_1\|=\cdots=\|x_n\|=1 and ∣T(x1,…,xn)∣=∥T∥|T(x_1, \ldots, x_n)|=\|T\|,
where L(nE){\mathcal L}(^n E) denotes the space of all continuous nn-linear forms on EE. For T∈L(nE),T\in {\mathcal L}(^n E), we define
Norm (T)={(x1,…,xn)∈En:(x1,…,xn) is a norming point of T}.\text{\rm Norm\,}(T) = \{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n) \ \text{is a norming point of}\ T\}.
Let Ro(w)2\mathbb{R}^2_{o(w)} denote R2\mathbb{R}^2 with the octagonal norm with weight 0<w≠10 < w\neq 1
∥(x,y)∥o(w)=max⁡{∣x∣+w∣y∣,∣y∣+w∣x∣}.\|(x, y)\|_{o(w)}=\max\big\{|x|+w|y|, |y|+w|x|\big\}.
We classify Norm (T)\text{\rm Norm\,}(T) for every T∈L(2Ro(w)2)T\in {\mathcal L}(^2 \mathbb{R}_{o(w)}^2) with weight 0<w≠10 < w\neq 1 in this paper.

Author information

Contact details are reproduced from the original publication and may be historical.

Sung Guen Kim

Dept. of Mathematics, Kyungpook National University, Daegu, Republic of Korea

sgk317@knu.ac.kr

Chang Yeol Lee

Dept. of Mathematics, Kyungpook National University, Daegu, Republic of Korea

Ukje Jeong

Dept. of Mathematics, Kyungpook National University, Daegu, Republic of Korea

Keywords

  • Norming points
  • bilinear forms

Mathematics Subject Classification

46A22

References

  1. R. M. Aron, C. Finet, E. Werner: Some remarks on norm-attaining n-linear forms, in: Function Spaces, Edwardsville 1994, Lecture Notes in Pure and Applied Mathematics 172, Dekker, New York (1995) 19–28.Google Scholar
  2. E. Bishop, R. Phelps: A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961) 97–98.https://doi.org/10.1090/S0002-9904-1961-10514-4Google Scholar
  3. Y. S. Choi, S. G. Kim: Norm or numerical radius attaining multilinear mappings and polynomials, J. London Math. Soc. (2) 54 (1996) 135–147.https://doi.org/10.1112/jlms/54.1.135Google Scholar
  4. S. Dineen: Complex Analysis on Infinite Dimensional Spaces, Springer, London (1999).https://doi.org/10.1007/978-1-4471-0869-6Google Scholar
  5. M. Jiménez Sevilla, R. Payá: Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math. 127 (1998) 99–112.https://doi.org/10.4064/sm-127-2-99-112Google Scholar
  6. S. G. Kim: The norming set of a polynomial in P(2 l∞ 2), Honam Math. J. 42/3 (2020) 569–576.Google Scholar
  7. S. G. Kim: The norming set of a bilinear form on l∞ 2, Comment. Math. 60/1-2 (2020) 37–63.Google Scholar
  8. S. G. Kim: The norming set of a symmetric 3-linear form on the plane with the l1 -norm, New Zealand J. Math. 51 (2021) 95–108.https://doi.org/10.53733/177Google Scholar
  9. S. G. Kim: The norming sets of L(2 l12) and Ls (2 l13), to appear in Bull. Trans. Univ. Brasov, Ser. III Math. Comput. Sci. 2(64) No. 2 (2022) 125–150.https://doi.org/10.31926/but.mif.2022.2.64.2.10Google Scholar

Suggested citation

S. G. Kim, C. Y. Lee, U. Jeong. “The Norming Set of a Bilinear Form on ℝ^2 with the Octagonal Norm.” Journal of Convex Analysis 30 (2023), No. 1, 111–130. https://doi.org/10.68381/jca30007

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press.