I am aware of the three theorems, let $X$ be a normed space:
i) $\overline{B}_{1}(0)^{*}$ is weak-* compact
ii) if $X$ is seperable the $\overline{B}_{1}(0)^{*}$ is weak-* sequentially compact
iii) if $X$ is Banach: $X$ Reflexive $\iff \overline{B}_{1}(0)$ weakly compact
This begs the question whether the same holds for any $r \neq 1>0$, I assume it does by scaling the unit ball, but I am unsure. On a more general note:
Oftentimes in a proof we only look at the unit ball $B_{1}(0)$ around $0$, and then state that since it holds for $B_{1}(0)$ it holds for any ball centered at $x \in X$ and $ r >0$, that is it will hold for $B_{r}(x)$, because of translations and scaling. Can we always reduce a given problem to the unit ball? Or when does this not hold?