Norm attaining multilinear forms on ℝn with the l1-norm

Let n,m ∈ ℕ with n,m ≥ 2. For given unit vectors x1, ・ ・ ・ , xn of a real Banach space E, we define NA(L(nE))(x1, ・ ・ ・ , xn) = {T ∈ L(nE) : |T(x1, ・ ・ ・ , xn)| = ∥T∥ = 1}, where L(nE) 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(mℓn1 ))(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.