On the duals of Lebesgue-Bochner $L^ p$ spaces
HTML articles powered by AMS MathViewer
- 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 ${L^1}{(\mu )^*} \simeq {L^\infty }(\nu )$.References
- Michael Cambern and Peter Greim, The dual of a space of vector measures, Math. Z. 180 (1982), no. 3, 373–378. MR 664522, DOI 10.1007/BF01214177
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
- N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189 N. Dunford and J. J. Schwartz, Linear operators, Part I, Interscience, New York and London, 1967.
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
- H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279, DOI 10.1007/978-3-642-65762-7
- J. Schwartz, A note on the space $L_p^*$, Proc. Amer. Math. Soc. 2 (1951), 270–275. MR 40588, DOI 10.1090/S0002-9939-1951-0040588-5
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