The Norming Sets of $$\mathcal{L}\left({}^{m}{l}_{1}^{n}\right)$$

Abstract

Let n ∈ ℕ, n ≥ 2. An element (x₁,…,x_n) ∈ E_n is called a norming point of T ∈ \(\mathcal{L}\left({}^{n}E\right)\) if ||x₁|| = … = ||x_n|| = 1 and |T(x₁,…,x_n)| = ||T||, where ℒ(ⁿE) denotes the space of all continuous n-linear forms on E. For T ∈ ℒ (ⁿE), we define

$$\text{Norm}\left(T\right)=\left\{\left({x}_{1},\dots ,{x}_{n}\right)\in {E}^{n}:\left({x}_{1},\dots ,{x}_{n}\right)\text{ is a norming point of }T\right\}.$$

The set Norm(T) is called the norming set of T. For m ∈ ℕ, m ≥ 2, we characterize Norm(T) for any T ∈ \(\mathcal{L}\left({}^{m}{l}_{1}^{n}\right)\), where \({l}_{1}^{n}={\mathbb{R}}^{n}\) with the l₁-norm. As applications, we classify Norm(T) for every T ∈ \(\mathcal{L}\left({}^{m}{l}_{1}^{n}\right)\) with n = 2, 3 and m = 2.