Norm attaining bilinear forms on the plane with the l1-norm

For given unit vectors x₁, · · ·, x_n of a real Banach space E, we define NA(ℒ(nE))(x1,…xn)={ T∈ℒ(nE):| T(x1,…xn) |=‖ T ‖=1 }, NA\left( {\mathcal{L}\left( {^nE} \right)} \right)\left( {{x_1}, \ldots {x_n}} \right) = \left\{ {T \in \mathcal{L}\left( {^nE} \right):\left| {T\left( {{x_1}, \ldots {x_n}} \right)} \right| = \left\| T \right\| = 1} \right\}, 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 classify NA(ℒ(²l₁²))((x₁, x₂), (y₁, y₂)) for unit vectors (x₁, x₂), (y₁, y₂)∈ l₁², where l₁² = ℝ² with the l₁-norm.