Given a Banach space $X$, are weak$^*$ bounded subsets of the dual space $X '$ also strongly bounded (with respect to the usual norm in $X '$)?

Question

Some related facts I already know:

1) In a Banach space $X$, weakly bounded sets are strongly bounded and vice-versa (Thm 3.18 - "Functional Analysis", Rudin);

2) From 1, it follows that my question is equivalent to proving that weak$^*$ bounded sets are bounded with respect to the weak topology of the dual $X'$.

3) If X is reflexive, then 2 is easy to show and then my question is true. But reflexiveness is really necessary?

Answer 1

Reflexivity is not necessary. Since $X$ is a Banach space, the Banach-Steinhaus theorem asserts that a family of continuous linear functionals on $X$ that is pointwise bounded (weak* bounded) is equicontinuous (norm-bounded).