TITLE: Dual space of $L^2(\mathbb{R},L^1(0,1))$?
QUESTION [11 upvotes]: I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a bit too quick, as there many references that suggest that this formally replacing each space by its dual is only possible in the reflexive case. 
Here, is already a good answer at Math.stackexchange, but it only works for finite measure spaces: click me.

REPLY [2 votes]: Your question has been partially answered in that it has been shown that the naive version is false. However, there is a concrete representation of the dual which might be of interest. The result becomes more transparent in a more general setting which might also be of interest to you.  Consider the space $L^p(\mu,E)$ where $\mu$ is a finite measure, $1\leq p<\infty$ and $E$ is a Banach space. Then its dual is naturally identifiable with $L^q(\mu,E')$ if and only if $E'$ has RNP ($q$ the conjugate of $p$). This is treated in detail in the beginning of chapter IV of "Vector measures" by Diestel and Uhl. However, there is a representation which works for any Banach space.  I have not found this result explicitly in the literature but the basic idea
is due to L. Schwartz (Sem. d'anal. fonctionelle (1974-75), Exp. 4---readily available online).  I will state the result for the case of separable $E$: The dual is the space $L^q_{w^\ast}(\mu,E')$, where the subscript $w\ast$ refers to the fact that we are using functions which are weak star rather than norm measurable. Thus in your case, the dual is $L^2_{w\ast}(\mu,L^\infty)$. (As pointed out above there are standard methods to extend such results to the case of $\sigma$-finite measures, in particular the real line with Lebesgue measure).