Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\Omega,X),\Vert\cdot\Vert_{p})$ to usual Bochner-Lebesgue spaces, i.e. $f:\Omega\to X$ measurable is in $L^{p}(\Omega,X)$ if and only if $$\Vert f\Vert_{p}=\bigg(\int_{\Omega}\Vert f(p)\Vert_{X}^{p} d\mu(x)\bigg)^{\frac{1}{p}}<\infty$$
Now, my question is the following:
Let $p\in (0,\infty)$ and $q$ such that $\frac{1}{p}+\frac{1}{q}=1$. Under which assumptions on $(\Omega,\mathcal{F},\mu)$ and $(X,\Vert\cdot\Vert_{X})$ is it true that $L^{p}(\Omega,X)^{\prime}\cong L^{q}(\Omega,X^{\prime})$?
where I denote by $\prime$ the topological dual space, i.e. the space of continuous linear functionals. Some remarks:
- If $X=\mathbb{R}$ or $\mathbb{C}$, then it is very well-known that the claim is true.
- A result in this direction can be found in Diestel, Uhl: Vector Measures, which states that the result is true if $(\Omega,\mathcal{F},\mu)$ is finite and if $X^{\prime}$ has the Radon-Nikodym property. An answer to this post on mathoverflow seems to quote that the same is true for $\sigma$-finite measure spaces, but this claim seems to be not contained in the book by Diestel-Uhl, if I am not mistaken.
I would greatly appreciate if someone has a reference on this where the corresponding isomorphism is stated explicitely.
As a follow-up question:
What about $L^{1}(\Omega,X)$ and $L^{\infty}(\Omega,X)$? Are there any necessary conditions on the measure and Banach space known in order for an isomorphism of the form $(L^{1}(\Omega,X))^{\prime}\cong L^{\infty}(\Omega,X^{\prime})$ to hold?
- I expect this this question to be much more subtle: In the easy case $X=\mathbb{R},\mathbb{C}$, the answer is positive if and only if $(\Omega,\mathcal{F},\mu)$ is localizable.
- Some results along this lines are proven in Ringström: The Cauchy Problem in General Relativity for Hilbert spaces. More explicitely, it is shown in Proposition 3.6 that the dual of $L^{1}([0,T],H)$ is isomorphic to $L^{\infty}([0,T],H)$ for a Hilbert space $H$.
