On the Bishop-Phelps-Bollobás theorem for multilinear mappings

We study the Bishop-Phelps-Bollobás property and the Bishop-Phelps-Bollobás property for numerical radius. Our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases. We characterize the pair (ℓ1(X),Y) to have the BPBp for bilinear forms and prove that on L1(μ) the numerical radius and the norm of a multilinear mapping are the same. We also show that L1(μ) fails the BPBp-nu for multilinear mappings although L1(μ) satisfies it in the operator case for every measure μ.