On weak compactness in $L^ 1(\mu , X)$
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- by J. Diestel, W. M. Ruess and W. Schachermayer PDF
- Proc. Amer. Math. Soc. 118 (1993), 447-453 Request permission
Abstract:
We present a characterization of weak compactness in ${L^1}(\mu ,X)$ and in more general Banach spaces of vector-valued measurable functions. Moreover, we slightly refine Talagrand’s parametrized version of Rosenthal’s ${l^1}$-theorem and extend it to ${L^1}(\mu ,X)$-bounded sequences.References
- Jürgen Batt, On weak compactness in spaces of vector-valued measures and Bochner integrable functions in connection with the Radon-Nikodým property of Banach spaces, Rev. Roumaine Math. Pures Appl. 19 (1974), 285–304. MR 341081 J. Diestel, Remarks on weak compactness in ${L^1}(\mu ,X)$, Glasgow Math. J. 18 (1977), 87-91.
- 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
- Jean Dieudonné, Sur les espaces de Köthe, J. Analyse Math. 1 (1951), 81–115 (French). MR 41347, DOI 10.1007/BF02790084 N. Dunford and J.T. Schwartz, Linear operators, Part 1, Interscience, New York, 1958.
- D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. MR 0454575, DOI 10.1017/CBO9780511897207
- A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168–186 (French). MR 47313, DOI 10.2307/2372076
- J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229. MR 210177, DOI 10.1007/BF02020976
- Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
- Michel Talagrand, Weak Cauchy sequences in $L^{1}(E)$, Amer. J. Math. 106 (1984), no. 3, 703–724. MR 745148, DOI 10.2307/2374292 A. Ülger, Weak compactness in ${L^1}(\mu ,X)$, Proc. Amer. Math. Soc. 113 (1991), 143-149.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 447-453
- MSC: Primary 46E40
- DOI: https://doi.org/10.1090/S0002-9939-1993-1132408-X
- MathSciNet review: 1132408