The norming sets of multilinear forms on R2 with a rotated supremum norm

Let n ∈ N, n ≥ 2. An element (x1,…, xn) ∈ En is called a norming point of T ∈ L(nE) if ∥x1∥ = … = ∥xn∥ = 1 and |T(x1,…, xn)| = ∥T∥, where L(nE) denotes the space of all continuous n-linear form on E. For T ∈ L(nE), we define Norm(T) = {(x1,…, xn) ∈ En : (x1…, xn) is a norming point of T}. Norm(T) is called the norming set of T. Let 0 ≤ θ < π/4 and ℓ2(∞,θ) = R2 with the rotated supremum norm ∥(x, y)∥(∞,θ) = max{|x cos θ + y sin θ|, |x sin θ − y cos θ|}. In this paper, we characterize Norm(T) for every T ∈ L(mℓ2(∞,θ)) for m ≥ 2.