A note on the topology in dual Banach spaces

If X is a Banach space, the γ topology of the dual X ∗ is the topology of the uniform convergence on bounded separable subsets of X . We prove, among other things, that co ¯ ( K ) = co ¯ γ ( K ) for every w ∗ -compact subset K of X ∗ . As an application we obtain some new separation results.