About $L^\infty$ in the dual of $L^1$

Question

Does someone know where can I find a proof of the following?

$$L^\infty(\mathbb{T}, X^*)\subseteq\left(L^1(\mathbb{T},X)\right)^*$$

(where $X^*$ denotes the dual space of $X$).

Answer 1

This duality holds if and only if $X^*$ has the Radon-Nikodym property with respect to the Haar measure on $\mathbb{T}$ which is equivalent to having the Radon-Nikodym property with respect to the Lebesgue measure on the unit interval. This is Theorem 1 on p. 98 in

J. Diestel and J.J. Uhl, Vector measures. Mathematical Surveys, Vol. 15, Amer. Math. Soc., Providence (1977).

As soon as $X^*$ has RNP (for example this is the case if $X$ is reflexive), you proceed like in this post.