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

Abstract

Abstract For given unit vectors x1, · · ·, xn 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, . . ., xn)|. 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.