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)



Banach spaces in which every compact lies inside the range of a vector measure

Authors: C. Piñeiro and L. Rodríguez-Piazza
Journal: Proc. Amer. Math. Soc. 114 (1992), 505-517
MSC: Primary 46B20; Secondary 46G10
MathSciNet review: 1086342
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the compact subsets of a Banach space $ X$ lie inside ranges of $ X$-valued measures if and only if $ {X^*}$ can be embedded in an $ {L^1}$ space. In these spaces we prove that every compact is, in fact, a subset of a compact range. We also prove that if every compact of $ X$ is a subset of the range of an $ X$-valued measure of bounded variation, then $ X$ is finite dimensional. Thus we answer a question by R. Anantharaman and J. Diestel.

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

  • [AD] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, preprint. MR 1122692 (92g:46049)
  • [De] D. W. Dean, The equation $ L(E,{X^{**}}) = L{(E,X)^{**}}$ and the principle of local reflexity, Proc. Amer. Math. Soc. 40 (1973), 146-148. MR 0324383 (48:2735)
  • [D] J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math., vol. 92, Springer-Verlag, New York, 1984. MR 737004 (85i:46020)
  • [DU] J. Diestel and J. J. Uhl, Vector measures, Math. Surveys Monogr., vol. 15, Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
  • [KK] I. Kluvanek and G. Knowles, Vector measures and control systems, Math. Stud., vol. 20, North-Holland, 1976. MR 0499068 (58:17033)
  • [J] G. J. O. Jameson, Summing and nuclear norms in Banach space theory, Stud. Texts, vol. 8, London Math. Soc., Cambridge Univ. Press, 1987. MR 902804 (89c:46020)
  • [LP] J. Lindenstrauss and A. Pełczynski, Absolutely summing operators in $ {\mathcal{L}_p}$-spaces and their applications, Studia Math. 29 (1968), 275-326. MR 0231188 (37:6743)
  • [P] G. Pisier, Factorization of linear operators and geometry of Banach spaces, Conf. Board Math. Sci. Regional Conf. Ser. Math., vol. 60, Amer. Math. Soc., Providence, RI, 1984. MR 829919 (88a:47020)
  • [R] L. Rodriguez-Piazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), 205-214. MR 1025281 (91e:46053)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20, 46G10

Retrieve articles in all journals with MSC: 46B20, 46G10

Additional Information

Keywords: Vector measures, range, Banach spaces, compact sets, subspaces of $ {L^1}$
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society