Norm attaining bilinear forms of $$${\mathcal L}(^2 d_{*}(1, w)^2)$$$ at given vectors

S.G. Kim (Kyungpook National University),


For given unit vectors $$$x_1, \cdots, x_n$$$ of a real Banach space $$$E,$$$ we define $$NA({\mathcal L}(^nE))(x_1, \cdots, x_n)=\{T\in {\mathcal L}(^nE): |T(x_1, \cdots, x_n)|=\|T\|=1\},$$ where $$${\mathcal L}(^nE)$$$ denotes the Banach space of all continuous $$$n$$$-linear forms on $$$E$$$ endowed with the norm $$$\|T\|=\sup_{\|x_k\|=1, 1\leq k\leq n}{|T(x_1, \ldots, x_n)|}$$$.
In this paper, we classify $$$NA({\mathcal L}(^2 d_{*}(1, w)^2))(Z_1, Z_2)$$$ for unit vectors $$$Z_1, Z_2\in d_{*}(1, w)^2,$$$ where $$$d_{*}(1, w)^2=\mathbb{R}^2$$$ with the norm of weight $$$0<w<1$$$ endowed with $$$\|(x, y)\|_{d_*(1, w)}=\max\Big\{|x|, |y|, \frac{|x|+|y|}{1+w}\Big\}$$$.


norm attaining bilinear forms; the plane with an octagonal norm

MSC 2020


Full Text:



Aron R.M., Finet C., Werner E. "Some remarks on norm-attaining $$$n$$$-linear forms", Function spaces, Edwardsville, IL, 1994; Lecture Notes in Pure and Appl. Math., Dekker, 1995; 172: pp. 19-28.

Bishop E., Phelps R. "A proof that every Banach space is subreflexive", Bull. Amer. Math. Soc., 1961; 67: pp. 97-98. doi:10.1090/S0002-9904-1961-10514-4

Choi Y.S., Kim S.G. "Norm or numerical radius attaining multilinear mappings and polynomials", J. London Math. Soc., 1996; 54(1): pp. 135-147. doi:10.1112/jlms/54.1.135

Dineen S. Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, 1999.

Jimenez Sevilla M., Paya R. "Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces", Studia Math., 1998; 127: pp. 99-112.

Kim S.G. "Explicit norm attaining polynomials", Indian J. Pure Appl. Math., 2003; 34: pp. 523-527.

Kim S.G. "Extreme bilinear forms of $$${\mathcal L}(^2 d_{*}(1, w)^2)$$$", Kyungpook Math. J., 2013; 53: pp. 625-638.

Kim S.G. "Norm attaining bilinear forms on the plane with the $$$\ell_1$$$-norm", Acta Univ. Sapientiae Math., 2022; 14(1): pp. 115-124. doi:10.2478/ausm-2022-0008

Kim S.G. "Norm attaining multilinear forms on $$$\mathbb{R}^n$$$ with the $$$\ell_1$$$-norm", to appear in Bull. Transilv. Univ. Brasov, Ser. III: Math. Comput. Sci., 2023; 3(65) (2).




  • There are currently no refbacks.

Copyright (c) 2023 S.G. Kim

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991