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Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p(X, \mu, E)$. I read somewhere that

If $E$ is reflexive, then $L_p(X, \mu, E)$ is reflexive and $(L_p(X, \mu, E))^* = L_q(X, \mu, E^*)$ with $1/p+1/q=1$.

I proved these claims for a particular case of Hilbert spaces here and here. My proofs depend on the inner product and thus can not be generalized to general reflexive Banach spaces.

Could you provide a reference for the proofs of these results?

Akira
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    See Diestel & Uhl, chapter 4. They characterize the dual spaces of $L^p(\mu, E)$. Proof works with Radon-Nikodym property. – daw Aug 17 '22 at 05:54
  • @daw It seems to me that (in the book) $E$ is reflexive $\implies$ $E$ has RNP $\implies$ $(L_p(E))^* = L_q(E^*)$. I could not find the proof that $E$ is reflexive $\implies$ $L_p (E)$ is reflexive. – Akira Aug 17 '22 at 06:19
  • @Masacroso I only found the duality when $E \in {\mathbb R, \mathbb C}$ at page 206. Could you elaborate more? – Akira Aug 17 '22 at 07:17
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    You could just use the same theorem in Diestel & Uhl (sec 4.1) twice: $(L^p(E))^{} = (L^q(E^{}))^ = L^p(E^{}) = L^p(E)$ since both $E$ and $E^*$ are reflexive, so have RNP. The following corollary in the book gives exactly the result you're after. – Onur Oktay Aug 17 '22 at 13:16
  • Thank you so much @OnurOktay. Could you post your comment as an answer? – Akira Aug 18 '22 at 01:09

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