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Let ${\rm S}^{n-1}$ be the unit sphere of ${\bf R}^n$ and let us consider the dual of the space $L^p\left({\bf R}^n; C^1({\rm S}^{n-1})\right)$, for some $p\in\langle1,\infty\rangle$: it is the space of all weakly-$\ast$ measurable functions $T:{\bf R}^n\to \left(C^1({\rm S}^{n-1})\right)'$ such that the integral $\int_{{\bf R}^{n}} \| T(x)\|_{\left(C^1({\rm S}^{n-1})\right)'}\; dx$ is finite. It is sometimes denoted by $L^{p'}_w\left({\bf R}^n;\left(C^1({\rm S}^{n-1})\right)'\right)$ (see the proof on the page 33 of click here).

My question is: does (at least in some form) the Radon-Nikodym property (click here) hold for space $L^{p'}_w\left({\bf R}^n;\left(C^1({\rm S}^{n-1})\right)'\right)$?

I have seen several results that require reflexivity, but $C^1({\rm S}^{n-1})$ is not reflexive (we only have separability), so they do not apply to this case.

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  • $\begingroup$ For the classical Radon-Nikodym property, the answer is negative. In the book Vector measures by Diestel and Uhl , at the bottom of page 98 there is Theorem 1 which gives a characterisation of spaces with RNP. If the RNP were to hold, then we would have that the dual of $L^p({\bf R}^d; C^1({\rm S}^{d-1}))$ would be precisely $L^{p'}\left({\bf R}^d; (C^1({\rm S}^{d-1}))'\right)$, which is not the case. $\endgroup$
    – Semmel
    Commented Feb 18, 2017 at 12:23

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