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Representation theorems of the dual of Lebesgue-Bochner function spaces. (English) Zbl 0959.46024

Let \((B,|\cdot|)\) be a Banach space with dual space \(B'\), and let \((\Omega,{\mathcal A},\mu)\) be a positive measure space. The Banach space \(B\) is said to have a Radon-Nikodým Property (RNP) with respect to \((\Omega,{\mathcal A},\mu)\) if \(B\) has the RNP with respect to every finite measure space \((A,A\cap{\mathcal A},\mu_A)\) with \(\mu(A)< \infty\). \(L^p(\mu, B)\) denotes the space of \(B\)-valued functions \(f\) with \(\|f\|_p= \{\int_\Omega|f|^p d\mu\}^{1/p}< \infty\).
The main results of this paper include statements identifying the dual space \((L^p(\mu, B))'\) with \(L^{p'}(\mu, B')\), where \(1< p<\infty\), \((1/p)+ (1/p')= 1\), if and only if \(B'\) has RNP with respect to \((\Omega,{\mathcal A},\mu)\).
The paper includes references to earlier publications in which conditions on the measure \(\mu\) are required for deriving the representation of the dual spaces. In particular, it is indicated that the results of this paper are based on results stated by the author in Sci. China, Ser. A 39, No. 10, 1034-1041 (1996; Zbl 0868.46014) and J. Xiamen Univ., Nat. Sci. 36, No. 4, 499-502 (1997; Zbl 0902.46053).

MSC:

46E40 Spaces of vector- and operator-valued functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
28B05 Vector-valued set functions, measures and integrals
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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