Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space. A countably additive maps $\mu:\Sigma\to A$ is called a vector measure. For any vector measure $\mu$ the semi-variation is defined by $$ ||\mu|| = \sup \left| \sum_i x_i\mu(E_i)\right|, $$ where the supremum extends over all partitions of $S$ into finite number of disjoint sets $\{E_i\}\subset\Sigma$ and all functionals $\{x_i\}\subset A^*$ (the dual of $A$).

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