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

Piñeiro, C.; Rodríguez-Piazza, L.

doi:10.1090/S0002-9939-1992-1086342-3

Banach spaces in which every compact lies inside the range of a vector measure
HTML articles powered by AMS MathViewer

by C. Piñeiro and L. Rodríguez-Piazza

Proc. Amer. Math. Soc. 114 (1992), 505-517

DOI: https://doi.org/10.1090/S0002-9939-1992-1086342-3

PDF | Request permission

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

R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), no. 2, 221–235. MR 1122692
David W. Dean, The equation $L(E,\,X^{\ast \ast })=L(E,\,X)^{\ast \ast }$ and the principle of local reflexivity, Proc. Amer. Math. Soc. 40 (1973), 146–148. MR 324383, DOI 10.1090/S0002-9939-1973-0324383-X
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
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
Igor Kluvánek and Greg Knowles, Vector measures and control systems, North-Holland Mathematics Studies, Vol. 20, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. MR 0499068
G. J. O. Jameson, Summing and nuclear norms in Banach space theory, London Mathematical Society Student Texts, vol. 8, Cambridge University Press, Cambridge, 1987. MR 902804, DOI 10.1017/CBO9780511569166
J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 829919, DOI 10.1090/cbms/060
Luis Rodríguez-Piazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), no. 1, 205–214. MR 1025281, DOI 10.1090/S0002-9939-1991-1025281-X