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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?

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  • $\begingroup$ I think you are missing the condition $\|x_i\|\le1$ for all $i$. Also, you should provide a reference to the definition of the semi-variation. $\endgroup$
    – Iosif Pinelis
    May 17, 2023 at 13:21
  • $\begingroup$ Thanks. I corrected the definition of the semi-variation. $\endgroup$
    – user72829
    May 17, 2023 at 13:32
  • $\begingroup$ How come $x\mu$ is a vector measure? Also, is the vector space $A$ over $\mathbb C$? $\endgroup$
    – Iosif Pinelis
    May 17, 2023 at 13:42
  • $\begingroup$ Sorry, I meant $x\mu$ is a complex measure. $\endgroup$
    – user72829
    May 17, 2023 at 13:43

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