Dual of $\ell_\infty(X)$

Given a Banach space $X$. Consider the space $\ell_\infty(X)$ which is the $\ell_\infty$-sum of countably many copies of $X$. Is there any accessible respresentation of the dual space $\ell_\infty(X)^*$? In particular, is this dual space isomorphic to the space of finitely additive $X^*$-valued measures on the powerset of $\mathbb N$ equipped with the semivariation norm?

Any references will be appreciated.

-
Is $E = X$ in your question? –  Christopher A. Wong Nov 18 '12 at 21:23
Oh yes, sorry. I'll correct it. –  Slavoj Žižek Nov 18 '12 at 21:36

There is no good description of the dual of $\ell_\infty(X)$ as far as I know. If $X$ is finite dimensional, then the answer to your second question is yes. Otherwise, it is no, for there is no way to define an action of a finitely additive $X^*$-valued measure on $\ell_\infty(X)$ if the ball of $X$ is not compact.

-
Suppose $c_{0}$ is the subspace of $l^{\infty}(X)$ consisting of all $x$ for which $x_{i}\rightarrow 0$ as $i\rightarrow 0$. Then $c_{0}^{*}=l^{1}$. This means to every bounded linear functional $\Lambda$ on $c_{0}$ there corresponds a unique sequence $\{\delta_{i}\}$ such that $$\sum |\delta_{i}|=|\Lambda|,\Lambda x=\sum x_{i}\delta_{i}$$
The reason $(l^{\infty})^{*}\not=l^{1}$ depends on the fact that there is a nontrivial bound linear function on $l^{\infty}$ which vanishes on all of $c_{0}$.
Note that in the question $X$ is a Banach space and $\ell^{\infty}(X)$ is the space of bounded functions $\mathbb{N} \to X$ so (unless I am misreading) you only seem to be addressing the case $X = \mathbb{R}$ which the OP is trying to generalize. In the case $X = \mathbb{R}$, the dual of $\ell^\infty$ can be described in terms of finitely additive measures and the OP asks whether a similar interpretation holds true for $X$ an arbitrary Banach space. –  Martin Jan 9 '13 at 7:05
I see. I checked Rudin again and it assumed $X$ is either $\mathbb{R}$ or $\mathbb{C}$. –  Bombyx mori Jan 9 '13 at 7:24