The Bishop-Phelps-Bollobás Theorem for bilinear forms

Abstract

In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space Y Y satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on ℓ 1 × Y \ell _1 \times Y is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space C ( K ) \mathcal {C}(K) of continuous functions on a compact Hausdorff topological space K K and the space K ( H ) K(H) of compact operators on a Hilbert space H H . On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on ℓ 1 × L 1 ( μ ) \ell _1 \times L_1 (\mu ) fails for any infinite-dimensional L 1 ( μ ) L_1 (\mu ) , a result that was known only when L 1 ( μ ) = ℓ 1 L_1 (\mu ) = \ell _1 .