The norming set of a symmetric bilinear form on the plane with the supremum norm

S. G. Kim

doi:10.30970/ms.55.2.171-180

S. G. Kim Department of Mathematics, Kyungpook National University, Republic of Korea

DOI: https://doi.org/10.30970/ms.55.2.171-180

Keywords: norming points;, symmetric bilinear forms

Abstract

An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}_s(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and
$|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}_s(^n E)$ denotes the space of all symmetric continuous $n$-linear forms on $E.$
For $T\in {\mathcal L}_s(^n E),$ we define $$\mathop{\rm Norm}(T)=\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\}.$$
$\mathop{\rm Norm}(T)$ is called the {\em norming set} of $T$. We classify $\mathop{\rm Norm}(T)$ for every $T\in {\mathcal L}_s(^2l_{\infty}^2)$.

References

R.M. Aron, C. Finet, E. Werner, Some remarks on norm-attaining n-linear forms, Function spaces (Edwardsville, IL, 1994), 19–28, Lecture Notes in Pure and Appl. Math., V. 172, Dekker, New York,1995.

E. Bishop, R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67 (1961), 97–98.

Y.S. Choi, S.G. Kim, Norm or numerical radius attaining multilinear mappings and polynomials, J. London Math. Soc., 54 (1996), №2, 135–147.

Y.S. Choi, S.G. Kim, The unit ball of $mathcal{P}(^2l_2^2)$, Arch. Math., Basel, 71 (1998), 472–480.

S. Dineen, Complex analysis on infinite dimensional spaces, Springer-Verlag, London, 1999.

M. Jimenez Sevilla, R. Paya, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math., 127 (1998), 99–112.

S.G. Kim, The unit ball of ${mathcal L}_s(^2l_{infty}^2$), Extracta Math., 24 (2009), 17–29.

S.G. Kim, The norming set of a polynomial in ${mathcal P}(^2 l_{infty}^2)$; Honam Math. J., 42 (2020), №3, 569–576.