TY - JOUR
T1 - NORM ATTAINING MULTILINEAR FORMS ON Rn WITH THE ℓ1-NORM
AU - Kim, Sung Guen
N1 - Publisher Copyright:
© 2024, Transilvania University of Brasov 1. All rights reserved.
PY - 2024/9/3
Y1 - 2024/9/3
N2 - Let n, m ∈ N with n, m ≥ 2. For given unit vectors x1, · · ·, xn of a real Banach space E, we define NA(L(n E))(x1, · · ·, xn) = {T ∈ L(n E): |T (x1, · · ·, xn)| = ∥T ∥ = 1}, where L(n E) denotes the Banach space of all continuous n-linear forms on E endowed with the norm ∥T ∥ = sup∥xk ∥=1,1≤k≤n |T (x1, …, xn)|. In this paper, we present a characterization of the elements in the set NA(L(m ℓn1))(W1, · · ·, Wm) for any given unit vectors W1, …, Wm ∈ ℓn1, where ℓn1 = Rn with the ℓ1-norm. This result generalizes the results from [7], and two particular cases for it are presented in full detail: the case n = 2, m = 2, and the case n = 3, m = 2.
AB - Let n, m ∈ N with n, m ≥ 2. For given unit vectors x1, · · ·, xn of a real Banach space E, we define NA(L(n E))(x1, · · ·, xn) = {T ∈ L(n E): |T (x1, · · ·, xn)| = ∥T ∥ = 1}, where L(n E) denotes the Banach space of all continuous n-linear forms on E endowed with the norm ∥T ∥ = sup∥xk ∥=1,1≤k≤n |T (x1, …, xn)|. In this paper, we present a characterization of the elements in the set NA(L(m ℓn1))(W1, · · ·, Wm) for any given unit vectors W1, …, Wm ∈ ℓn1, where ℓn1 = Rn with the ℓ1-norm. This result generalizes the results from [7], and two particular cases for it are presented in full detail: the case n = 2, m = 2, and the case n = 3, m = 2.
KW - norm attaining multilinear forms
KW - ℓ1
UR - http://www.scopus.com/inward/record.url?scp=85204242770&partnerID=8YFLogxK
U2 - 10.31926/but.mif.2024.4.66.2.10
DO - 10.31926/but.mif.2024.4.66.2.10
M3 - Article
AN - SCOPUS:85204242770
SN - 2810-2029
VL - 4
SP - 173
EP - 184
JO - Bulletin of the Transilvania University of Brasov, Series III: Mathematics and Computer Science
JF - Bulletin of the Transilvania University of Brasov, Series III: Mathematics and Computer Science
IS - 2
ER -