On convergence on the boundary of the unit ball in dual space

In this paper some results that are known for extreme points of the unit ball in dual space are carried over to a more general case, namely to the case of the boundary of the ball ($\Gamma\subset B$ is the boundary of the unit ball $B$ in the space dual to $X$ if every $x\in X$ achieves its maximum value on $B$ at some point of $\Gamma$). For example, it is established that if a set is bounded in $X$ and countably compact in $\sigma(X,\Gamma)$, then it is weakly compact in $X$.