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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the duals of Lebesgue-Bochner $L^ p$ spaces


Author: Bahattin Cengiz
Journal: Proc. Amer. Math. Soc. 114 (1992), 923-926
MSC: Primary 46E40
DOI: https://doi.org/10.1090/S0002-9939-1992-1027088-7
MathSciNet review: 1027088
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Abstract: Let $(X,\mathcal {A},\mu )$ be an arbitrary positive measure space. We prove that there exist an extremally disconnected (locally) compact Hausdorff space $Y$ and a perfect (regular) Borel measure $\nu$ on $Y$ such that ${L^p}(\mu ,\textrm {E}) \simeq {L^p}(\nu ,E)$ for all $1 \leq p < \infty$ and any Banach space $E$. If ${E^*}$ is separable, then ${L^p}(\mu ,\textrm {E})* \simeq {L^q}(\mu ,{\textrm {E}^*})$ for all $1 < p < \infty ,\;\frac {1}{p} + \frac {1}{q} = 1$ , and ${L^1}(\mu ,\textrm {E})* \simeq {L^\infty }(\nu ,{\textrm {E}^*}) \simeq C(\beta Y,\textrm {E}_*^*)$, where $E_*^*$ denotes ${E^*}$ endowed with the weak* topology. In particular ${L^1}{(\mu )^*} \simeq {L^\infty }(\nu )$.


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Article copyright: © Copyright 1992 American Mathematical Society