The norming sets of ℒ( 2ℓ1 2) and ℒS( 2ℓ1 3)

Abstract

Let n ∈ N. An element (x₁, . . . , x_n) ∈ E ⁿ is called a norming point of T ∈ ℒ ( ⁿE) 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

Norm(T) = { n (x₁, . . . , x_n) ∈ En : (x₁, . . . , x_n) is a norming point of T } .

Norm(T) is called the norming set of T. We classify Norm(T) for every T ∈ ℒ ( ²ℓ₁²) or ℒ_s ( ²ℓ₁³), where ℓ₁ⁿ = ℝⁿ with the ℓ₁-norm.