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.