Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space. A countably additive map $\mu:\Sigma\to A$ is called a vector measure. For any vector measure $\mu$ the semi-variation is defined by
$$
||\mu|| = \sup \left\{ |x\mu|(S)\,|\,x\in A^*,\,\|x\|\leq 1\right\},
$$ 
where $A^*$ is the dual of $A$ and $|x\mu|$ is the total variation of the complex measure $x\mu$ (for details see e.g. <https://encyclopediaofmath.org/wiki/Vector_measure>). 

Is it true that $||\mu||$ is a norm in the space of vector measures? <br>
If yes, is the space of vector measures equipped with this norm complete?