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)

 
 

 

On weak compactness in Lebesgue-Bochner spaces


Author: José Rodríguez
Journal: Proc. Amer. Math. Soc. 144 (2016), 103-108
MSC (2010): Primary 46B50, 46G10
DOI: https://doi.org/10.1090/proc/12846
Published electronically: August 5, 2015
MathSciNet review: 3415580
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a Banach space, $ (\Omega ,\Sigma ,\mu )$ a probability space and $ K$ a weakly compact subset of $ L^p(\mu ,X)$, $ 1\leq p<\infty $. The following question was posed by J. Diestel: is there a weakly compactly generated subspace $ Y \subset X$ such that $ K \subset L^p(\mu ,Y)$? We show that, in general, the answer is negative. We also prove that the answer is affirmative if either $ \mu $ is separable or $ X$ is weakly sequentially complete.


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

  • [1] S. A. Argyros, Weakly Lindelöf determined Banach spaces not containing $ \ell ^1(N)$, preprint (1993), arXiv:math/9210210v1.
  • [2] Spiros A. Argyros and Sophocles Mercourakis, Examples concerning heredity problems of WCG Banach spaces, Proc. Amer. Math. Soc. 133 (2005), no. 3, 773-785 (electronic). MR 2113927 (2005m:46018), https://doi.org/10.1090/S0002-9939-04-07532-X
  • [3] Antonio Avilés, The number of weakly compact sets which generate a Banach space, Israel J. Math. 159 (2007), 189-204. MR 2342477 (2008m:46023), https://doi.org/10.1007/s11856-007-0042-6
  • [4] Jürgen Batt and Wolfgang Hiermeyer, On compactness in $ L_{p}(\mu ,\,X)$ in the weak topology and in the topology $ \sigma (L_{p}(\mu ,\,X),\,L_{q}(\mu ,\,X^{\prime } ))$, Math. Z. 182 (1983), no. 3, 409-423. MR 696537 (84m:46039), https://doi.org/10.1007/BF01179760
  • [5] J. Bourgain, An averaging result for $ l^{1}$-sequences and applications to weakly conditionally compact sets in $ L^{1}_{X}$, Israel J. Math. 32 (1979), no. 4, 289-298. MR 571083 (81i:46021), https://doi.org/10.1007/BF02760458
  • [6] Joe Diestel, Some problems arising in connection with the theory of vector measures, Séminaire Choquet, 17e année (1977/78), Initiation à l'analyse, Fasc. 2, Secrétariat Math., Paris, 1978, pp. Exp. No. 23, 11. MR 522987 (80d:46077)
  • [7] J. Diestel, W. M. Ruess, and W. Schachermayer, On weak compactness in $ L^1(\mu , X)$, Proc. Amer. Math. Soc. 118 (1993), no. 2, 447-453. MR 1132408 (93g:46033), https://doi.org/10.2307/2160321
  • [8] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964 (56 #12216)
  • [9] Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. The basis for linear and nonlinear analysis. MR 2766381 (2012h:46001)
  • [10] D. H. Fremlin, Measure theory. Vol. 2, Torres Fremlin, Colchester, 2003. Broad foundations; Corrected second printing of the 2001 original. MR 2462280 (2011a:28001)
  • [11] Richard Haydon, Nonseparable Banach spaces, Functional analysis: surveys and recent results, II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979) Notas Mat., vol. 68, North-Holland, Amsterdam-New York, 1980, pp. 19-30. MR 565396 (81e:46016)
  • [12] H. Elton Lacey, The isometric theory of classical Banach spaces, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 208. MR 0493279 (58 #12308)
  • [13] Michel Talagrand, Weak Cauchy sequences in $ L^{1}(E)$, Amer. J. Math. 106 (1984), no. 3, 703-724. MR 745148 (85j:46062), https://doi.org/10.2307/2374292
  • [14] A. Ülger, Weak compactness in $ L^1(\mu ,X)$, Proc. Amer. Math. Soc. 113 (1991), no. 1, 143-149. MR 1070533 (92g:46035), https://doi.org/10.2307/2048450

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B50, 46G10

Retrieve articles in all journals with MSC (2010): 46B50, 46G10


Additional Information

José Rodríguez
Affiliation: Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100 Espinardo (Murcia), Spain
Email: joserr@um.es

DOI: https://doi.org/10.1090/proc/12846
Keywords: Weakly compact set, weakly compactly generated Banach space, Lebesgue-Bochner space, separable measure
Received by editor(s): October 28, 2014
Published electronically: August 5, 2015
Additional Notes: This research was supported by MINECO and FEDER under project MTM2011-25377
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society