# Banach-Krein-Šmulian theorem

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 $r>0$.

## References

• 1 Dunford, N., and J. T. Schwartz, Linear Operators, Part I, Interscience Publishers, 1967.
Title Banach-Krein-Šmulian theorem BanachKreinvSmulianTheorem 2013-03-22 15:14:52 2013-03-22 15:14:52 georgiosl (7242) georgiosl (7242) 21 georgiosl (7242) Theorem msc 46H05