Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



$ L\sp{1}\sb{x}$ is weakly compactly generated if $ X$ is

Author: J. Diestel
Journal: Proc. Amer. Math. Soc. 48 (1975), 508-510
MSC: Primary 46E40; Secondary 46B05
MathSciNet review: 0367651
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Though good criteria for weak compactness in the space of Bochner-integrable functions are not yet known, one can show that $ {L_1}(\mu ;X)$ is a weakly compactly generated Banach space whenever $ \mu $ is finite and $ X$ is a weakly compactly generated Banach space. The proof depends upon a recent factorization scheme due to W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński.

References [Enhancements On Off] (What's this?)

  • [1] J. 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. (to appear). MR 0341081 (49:5831)
  • [2] J. K. Brooks and N. Dinculeanu, Strong additivity, absolute continuity and compactness in spaces of measures, J. Math. Anal. Appl. (to appear). MR 0346505 (49:11230)
  • [3] S. D. Chatterji, Weak convergence in certain special Banach spaces, MRC Technical Summary Report 443, University of Wisconsin, Madison, Wis., 1963.
  • [4] W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 411-420. MR 0355536 (50:8010)
  • [5] J. Diestel and B. Faires, Some remarks on the classical Banach operator ideals, Proc. Amer. Math. Soc. (to appear). MR 0454701 (56:12950)
  • [6] J. Diestel and J. J. Uhl, Jr., The Radon-Nikodým theorem for Banach space valued measures, Rocky Mountain J. Math. (to appear). MR 0399852 (53:3694)
  • [7] N. Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323-392. MR 1, 338. MR 0002020 (1:338b)
  • [8] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955). MR 17, 763. MR 0075539 (17:763c)
  • [9] R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516-541. MR 2, 318. MR 0004094 (2:318c)
  • [10] C. Swartz, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E40, 46B05

Retrieve articles in all journals with MSC: 46E40, 46B05

Additional Information

Keywords: Vector measures, weakly compact operators, Radon-Nikodým property, topological tensor products, weakly compactly generated spaces
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society