Let $E$ be a Banach space. A convex subset $C$ of the dual space $E^*$ is closed in the weak-$*$ topology if and only if the intersection of $C$ with the ball $B_{r}(0)$ is weak-$*$ closed for every $r>0$

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Dunford, N., and J. T. Schwartz, Linear Operators, Part I, Interscience Publishers, 1967.