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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the duals of Lebesgue-Bochner $L^ p$ spaces
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by Bahattin Cengiz PDF
Proc. Amer. Math. Soc. 114 (1992), 923-926 Request permission

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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Additional Information
  • © Copyright 1992 American Mathematical Society
  • 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