Geodesic
On convergence on the boundary of the unit ball in dual space
Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 753-758

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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 $X$. 
@article{MZM_1996_59_5_a10,
     author = {V. I. Rybakov},
     title = {On convergence on the boundary of the unit ball in dual space},
     journal = {Matemati\v{c}eskie zametki},
     pages = {753--758},
     publisher = {mathdoc},
     volume = {59},
     number = {5},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a10/}
}
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V. I. Rybakov. On convergence on the boundary of the unit ball in dual space. Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 753-758. http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a10/