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 :)