An extension of James's compactness theorem

Let X and Y be Banach spaces and F⊂BY⁎. Endow Y with the topology τF of pointwise convergence on F. Assume that T:X⁎→Y is a bounded linear operator which is (w⁎,τF) continuous. Assume further that every vector in the range of T attains its norm at some element of F (that is, for every x⁎∈X⁎ there exists y⁎∈F such that ‖T(x⁎)‖=|y⁎(Tx⁎)|). Then T is (w⁎,w) continuous. The proof relies on Rosenthal's ℓ1-theorem. As a corollary to the above result, one obtains an alternative proof of James's compactness theorem that a bounded subset K of a Banach space E is relatively weakly compact provided that each functional in E⁎ attains its supremum on K.