A subset of the norm dual of a normed space is weak* compact if and only if it is weak* closed and norm bounded.
This is stated without a proof in:
Infinite Dimensional Analysis A Hitchhiker's Guide
Authors: Aliprantis, Charalambos D., Border, Kim Page 235 6.21.
I know that this is true for Banach spaces and one direction from weak* closed and norm bounded to weak* compact is also true. But I´m not sure if a subset of of the norm dual which is weak* compact is also norm bounded.
If not does exist a counterexample?