Let $ X $ be Banach space, and $X^*$ its dual. A set $ F \subset X ^ * $ is weakly-* compact if and only if $ F $ is closed in the weak* topology and is bounded in norm.
How does one prove this (well-known) fact?
This characterization of compactness in weak* topology is similar to compactness in finite dimensional spaces, where it is equivalent to being closed and bounded.
The analogous statement for weak topology is false. An example is given in Weakly compact implies bounded in norm: the closed unit ball of $c_0$ is weakly closed and norm bounded, but not weakly compact.