Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space 
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. My question is, what is the dual $E'$ of $E$? Is there an identification as in the case of $L^{\infty}(\Omega,\mu)$ and $\ell^{\infty}$ by the mean of bounded additively functions of bounded variations? Is there such an duality result? If so can also give references for that?


Thank you very much :)



Duality of Bochner $L^{\infty}$ space
I take it from your question and comments that it is important that one considers the Banach space of bounded meaurable functions, not equivalence classes thereof. In the scalar case, the dual is the space of finitely additive bounded measures. In the vector valued case, the dual is, at least for the case of a separable Banach space, naturally identifiable with the space of bounded, finitely additive measures with values in the dual of $E$. The case of measurable functions with values in a non-separable Banach space is scary, at least for me.The classic "Vector measures" by Diestel and Uhl is probably still the best reference for such topics---an earlier classic is that of Dunford and Schwatrz.


