If we consider the dual Banach space $V' :=\{f:V\rightarrow \mathbb{C}$ such that $f$ is linear and bounded$\}$. We know $V'$ also forms a Banach space in the norm topology where the norm is the general operator norm. But the open ball is not compact in the norm topology (V is not finite dimensional), but it is so by the weak* topology by the Banach Alaoglu Theorem.
My question is that are these two topologies i.e. the norm topology and the weak* topology comparable, i.e. is one of them weaker than the other?
The second question is, whether $V'$ is still a Banach space with the weak* topology?