Banach spaces in which every compact lies inside the range of a vector measure
Authors:
C. Piñeiro and L. RodríguezPiazza
Journal:
Proc. Amer. Math. Soc. 114 (1992), 505517
MSC:
Primary 46B20; Secondary 46G10
MathSciNet review:
1086342
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Abstract: We prove that the compact subsets of a Banach space lie inside ranges of valued measures if and only if can be embedded in an 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 is a subset of the range of an valued measure of bounded variation, then is finite dimensional. Thus we answer a question by R. Anantharaman and J. Diestel.
 [AD]
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
(92g:46049)
 [De]
David
W. Dean, The equation
𝐿(𝐸,𝑋**)=𝐿(𝐸,𝑋)** and the
principle of local reflexivity, Proc. Amer.
Math. Soc. 40
(1973), 146–148. MR 0324383
(48 #2735), http://dx.doi.org/10.1090/S0002993919730324383X
 [D]
Joseph
Diestel, Sequences and series in Banach spaces, Graduate Texts
in Mathematics, vol. 92, SpringerVerlag, New York, 1984. MR 737004
(85i:46020)
 [DU]
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)
 [KK]
Igor
Kluvánek and Greg
Knowles, Vector measures and control systems, NorthHolland
Publishing Co., AmsterdamOxford; American Elsevier Publishing Co., Inc.,
New York, 1976. NorthHolland Mathematics Studies, Vol. 20; Notas de
Matemática, No. 58. [Notes on Mathematics, No. 58]. MR 0499068
(58 #17033)
 [J]
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
(89c:46020)
 [LP]
J.
Lindenstrauss and A.
Pełczyński, Absolutely summing operators in
𝐿_{𝑝}spaces and their applications, Studia Math.
29 (1968), 275–326. MR 0231188
(37 #6743)
 [P]
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
(88a:47020)
 [R]
Luis
RodríguezPiazza, The range of a vector measure
determines its total variation, Proc. Amer.
Math. Soc. 111 (1991), no. 1, 205–214. MR 1025281
(91e:46053), http://dx.doi.org/10.1090/S0002993919911025281X
 [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 and the principle of local reflexity, Proc. Amer. Math. Soc. 40 (1973), 146148. MR 0324383 (48:2735)
 [D]
 J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math., vol. 92, SpringerVerlag, 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, NorthHolland, 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 spaces and their applications, Studia Math. 29 (1968), 275326. 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. RodriguezPiazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), 205214. MR 1025281 (91e:46053)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199210863423
PII:
S 00029939(1992)10863423
Keywords:
Vector measures,
range,
Banach spaces,
compact sets,
subspaces of
Article copyright:
© Copyright 1992
American Mathematical Society
