# Semivariation of a vector measure in dual space

We define the semivariation of a vector measure $$\mu:\mathcal{B}(\mathbb{T})\to E$$ as

$$\|{\mu}\|(A)=\sup\lbrace{|\langle e^*,\mu\rangle|(A),\; e^*\in E^*,\;\|e^*\|=1\rbrace}$$

where $$\langle e^*,\mu\rangle(A)=e^*(\mu(A))$$, and $$|\cdot|$$ is the total variation for scalar measures.

I have read that in the case in which $$E$$ is a dual space, $$E=F^*$$, one can obtain this semivariation also in the following manner:

$$\|{\mu}\|(A)=\sup\lbrace{|\langle \mu,f\rangle|(A),\; f\in F,\;\|f\|=1\rbrace}$$

where $$\langle \mu,f\rangle(A)=\mu(A)(f)$$, and $$|\cdot|$$ is again the total variation for scalar measures.

How can one go from one definition to the other? Is it related with Hahn-Banach theorem?

• Wouldn't this follow from Goldstine's Theorem (A normed linear space $X$ is weak* dense in $X^{**}$)? – David Mitra Apr 15 at 11:13