NORM ATTAINING MULTILINEAR FORMS ON Rⁿ WITH THE ℓ₁-NORM

Let n, m ∈ N with n, m ≥ 2. For given unit vectors x₁, · · ·, x_n of a real Banach space E, we define NA(L(ⁿ E))(x₁, · · ·, x_n) = {T ∈ L(ⁿ E): |T (x₁, · · ·, x_n)| = ∥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, …, x_n)|. In this paper, we present a characterization of the elements in the set NA(L(^m ℓⁿ1))(W₁, · · ·, W_m) for any given unit vectors W₁, …, W_m ∈ 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.