Norm attaining multilinear forms on ℝn with the l1-norm

Abstract

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(ⁿE))(x₁, ・ ・ ・ , x_n) = {T ∈ L(ⁿE) : |T(x₁, ・ ・ ・ , x_n)| = ∥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, . . . , x_n)|. In this paper, we present a characterization of the elements in the set NA(L(^mℓⁿ₁ ))(W₁, ・ ・ ・ ,W_m) for any given unit vectors W₁, . . . ,W_m ∈ ℓⁿ₁ , where ℓⁿ₁ = ℝⁿ with the ℓ_1-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.