The Weak differentiability of norm and a generalized Bhatia-\v{S}emrl Theorem

We completely characterize the weak differentiability (or, in other words Gateaux differentiability) of the norm in the spaces of bounded multilinear maps. Also, we obtain a multilinear generalization of the well-known Bhatia-\v{S}emrl theorem on Birkhoff-James orthogonality.