The norming sets of L(mln1)
Abstract
UDC 517.9
Let n∈N, n≥2. An element (x1,…,xn)∈En is called a {\em norming point} of T∈L(nE) if\/ ‖x1‖=…=‖xn‖=1 and |T(x1,…,xn)|=‖T‖, where L(nE) denotes the space of all continuous n-linear forms on E. For T∈L(nE), we define Norm(T)={(x1,…,xn)∈En:(x1,…,xn)is a norming point ofT}. The Norm(T) is called the {\em norming set} of T. For m∈N, m≥2, we characterize Norm(T) for every T∈L(mln1), where ln1=Rn with the l1-norm. As applications, we classify Norm(T) for every T∈L(mln1) with n=2,3 and m=2.
References
R. M. Aron, C. Finet, E. Werner, Some remarks on norm-attaining n-linear forms, Lecture Notes in Pure and Appl. Math., 172, Function Spaces (Edwardsville, IL, 1994), Dekker, New York (1995), p. 19–28.
E. Bishop, R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67, 97–98 (1961). DOI: https://doi.org/10.1090/S0002-9904-1961-10514-4
Y. S. Choi, S. G. Kim, Norm or numerical radius attaining multilinear mappings and polynomials, J. London Math. Soc. (2), 54, 135–147 (1996). DOI: https://doi.org/10.1112/jlms/54.1.135
S. Dineen, Complex analysis on infinite dimensional spaces, Springer-Verlag, London (1999). DOI: https://doi.org/10.1007/978-1-4471-0869-6
M. Jim'enez Sevilla, R. Payá, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math., 127, 99–112 (1998). DOI: https://doi.org/10.4064/sm-127-2-99-112
S. G. Kim, The norming set of a bilinear form on l2∞, Comment. Math., 60, № 1-2, 37–63 (2020).
S. G. Kim, The norming set of a polynomial in P(2l2∞), Honam Math. J., 42, № 3, 569–576 (2020).
S. G. Kim, The norming set of a symmetric bilinear form on the plane with the supremum norm, Mat. Stud., 55, № 2, 171–180 (2021). DOI: https://doi.org/10.30970/ms.55.2.171-180
S. G. Kim, The norming set of a symmetric 3-linear form on the plane with the l1-norm, New Zealand J. Math., 51, 95–108 (2021). DOI: https://doi.org/10.53733/177
S. G. Kim, The norming sets of L(2l21) and Ls(2l31), Bull. Transilv. Univ. Brasov, Ser. III, 2(64), № 2, 125–150 (2022). DOI: https://doi.org/10.31926/but.mif.2022.2.64.2.10
S. G. Kim, The norming sets of L(2R2h(w)), Acta Sci. Math. (Szeged), 89, № 1-2, 61–79 (2023). DOI: https://doi.org/10.1007/s44146-023-00078-7
Copyright (c) 2024 Sung Guen Kim
This work is licensed under a Creative Commons Attribution 4.0 International License.