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?
If yes, is the space of vector measures equipped with this norm complete?

Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. 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).

Let $X$ be the vector space over the field of complex numbers of vector measures with finite semi-variation. It is easy to see that $||\mu||$ is a norm in $X$. Is $(X,||\cdot||)$ a Banach space?

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. If yes, is the space of vector measures equipped with this norm complete.

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?
If yes, is the space of vector measures equipped with this norm complete?

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 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.

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. If yes, is the space of vector measures equipped with this norm complete.