Sung Guen Kim Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

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Let n ∈ ℕ. An element (x1, … , xn) ∈ En is called a norming point of TLnE if x1==xn=1 and Tx1,,xn=T, where LnE denotes the space of all continuous symmetric n-linear forms on E. For TLnE, we define

NormT=x1,,xnEn:x1,,xn is a norming of T.

Norm(T) is called the norming set of T.

Let ·2 be the plane with a certain norm such that the set of the extreme points of its unit ball ext B·2=±W1,±W2 for some W1±W2·2.

In this paper, we classify Norm(T) for every TLn·2. We also present relations between the norming sets of Lnl2 and Lnl12.

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    ARON, R. M., FINET C., AND WERNER, E. Some remarks on norm-attaining n-linear forms, Function spaces (Edwardsville, IL, 1994), Lecture Notes in Pure and Appl. Math., 172, Dekker, New York, 1995, pp. 1928.

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    BISHOP, E. AND PHELPS, R. A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67, (1961), 9798.

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    CHOI, Y. S. AND kIM, S. G. Norm or numerical radius attaining multilinear mappings and polynomials, J. London Math. Soc. 54, 2 (1996), 135147.

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    DINEEN, S. Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London (1999).

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    JIMÉNEZ SEVILLA M. AND PAYÁ, R. Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math. 127, (1998), 99112.

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  • [6]

    KIM, S. G. The norming set of a bilinear form on l 2, Comment. Math. 60, (1–2) (2020), 3763.

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    KIM, S. G. The norming set of a polynomial in P 2 l 2, Honam Math. J. 42, 3 (2020), 569-576.

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    KIM, S. G. The norming set of a symmetric bilinear form on the plane with the supremum norm, Mat. Stud. 55, 2 (2021), 171180.

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    KIM, S. G. The norming set of a symmetric 3-linear form on the plane with the l1-norm, New Zealand J. Math. 51, (2021), 95108.

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    KIM, S. G. The norming sets of L 2 l 1 2  and  L s 2 l 1 3, Bull. Transilv. Univ. Brasov, Ser. III: Math. Comput. Sci. 64, 2 (2022), 125150.

  • [11]

    KIM, S. G. The norming sets of L 2 h w 2, to appear in Acta Sci. Math. (Szeged) 89, 34 (2023)

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