Remarks on the norming sets of L(mln1) and description of the norming sets of L(3l21)

  • Sung Guen Kim Department of Mathematics, Kyungpook National University Daegu, Republic of Korea
Keywords: norming points; multilinear forms on Rn with l1-norm

Abstract

Let nN,n2. An element x=(x1,,xn)En is called a {\em norming point} of TL(nE) if x1==xn=1 and
|T(x)|=T, where L(nE) denotes the space of all continuous n-linear forms on E.
For TL(nE) we define the {\em norming set} of T

Norm(T)={(x1,,xn)En:(x1,,xn) is a norming point of T}.

By i=(i1,i2,,im) we denote the multi-index. In this paper we show the following:

\noi (a) Let n,m2 and let ln1=Rn with the l1-norm. Let T=(ai)1iknL(mln1) with T=1.
Define S=(bi)1iknL(nlm1) be such that bi=ai if
|ai|=1 and bi=1 if
|ai|<1.

Let A={1,,n}××{1,,n} and M={iA:|ai|<1}.
Then,

Norm(T)=(i1,,im)M{((t(1)1,,t(1)i11,0,t(1)i1+1,,t(1)n),(t(2)1,,t(2)n),,(t(m)1,,t(m)n)),

((t(1)1,,t(1)n),(t(2)1,,t(2)i21,0,t(2)i2+1,,t(2)n),(t(3)1,,t(3)n),,(t(m)1,,t(m)n)),

,((t(1)1,,t(1)n),,(t(m1)1,,t(m1)n),(t(m)1,,t(m)im1,0,t(m)im+1,,t(m)n)):

((t(1)1,,t(1)n),,(t(m)1,,t(m)n))Norm(S)}.

This statement extend the results of [9].

\noi (b) Using the result (a), we describe the norming sets of every TL(3l21).

Author Biography

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

Department of Mathematics, Kyungpook National University
Daegu, Republic of Korea

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,

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

–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.

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

M.J. Sevilla, R. Pay´a, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence

spaces, Studia Math., 127 (1998), 99–112.

S.G. Kim, The norming set of a bilinear form on l2infty, Comment. Math., 60 (2020), №1–2, 37–63.

S.G. Kim, The norming set of a polynomial in mathcalP(2l2infty), Honam Math. J., 42 (2020), №3, 569–576.

S.G. Kim, The norming set of a symmetric bilinear form on the plane with the supremum norm, Mat.

Stud., 55 (2021), №2, 171–180.

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.

S.G. Kim, The norming sets of mathcalL(2l21) and mathcalLs(2l31),, to appear in Bull. Transilv. Univ. Brasov, Ser. III:

Math. Copmut. Sci., 2(64) (2022), №2.

S.G. Kim, The norming sets of mathcalL(2mathbbR2h(w)), to appear in Acta Sci. Math. (Szeged), 89 (2023), №1–2.

Published
2023-01-16
How to Cite
Kim, S. G. (2023). Remarks on the norming sets of L(mln1) and description of the norming sets of L(3l21). Matematychni Studii, 58(2), 201-211. https://doi.org/10.30970/ms.58.2.201-211
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