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 $$ ||\mu|| = \sup \left\{ |x\mu|(S)\,|\,x\in A^*\right\}, $$ where $A^*$ is the dual of $A$ and $|x\mu|$ is the total variation of the vector 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. If yes, is the space of vector measures equipped with this norm complete.