The Bishop-Phelps-Bollobás Theorem for bilinear forms

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 $Y$ satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on $\ell _1 \times Y$ is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space $\mathcal {C}(K)$ of continuous functions on a compact Hausdorff topological space $K$ and the space $K(H)$ of compact operators on a Hilbert space $H$. On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on $\ell _1 \times L_1 (\mu )$ fails for any infinite-dimensional $L_1 (\mu )$, a result that was known only when $L_1 (\mu ) = \ell _1$.