Previous |  Up |  Next


countably additive vector measure of bounded variation; Pettis integrable function space; copy of $c_{0}$; copy of $\ell _{\infty }$
If $(\Omega ,\Sigma ) $ is a measurable space and $X$ a Banach space, we provide sufficient conditions on $\Sigma $ and $X$ in order to guarantee that $\mathop {\mathrm bvca}( \Sigma ,X) $, the Banach space of all $X$-valued countably additive measures of bounded variation equipped with the variation norm, contains a copy of $c_{0}$ if and only if $X$ does.
[1] J. Bourgain: An averaging result for $c_{0}$-sequences. Bull. Soc. Math. Belg., Sér. B 30 (1978), 83–87. MR 0549653 | Zbl 0417.46019
[2] P. Cembranos, J.  Mendoza: Banach Spaces of Vector-Valued Functions. Lecture Notes in Mathematics Vol.  1676. Springer-Verlag, Berlin, 1997. MR 1489231
[3] J.  Diestel: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, 92, Springer-Verlag, New York-Heidelberg-Berlin, 1984. MR 0737004
[4] J.  Diestel, J.  Uhl: Vector Measures. Mathematical Surveys, No  15. Am. Math. Soc., Providence, 1977. MR 0453964
[5] L.  Drewnowski: When does $\mathop {\mathrm ca}( \Sigma ,Y) $ contain a copy of  $\ell _{\infty }$ or $c_{0}$. Proc. Am. Math. Soc. 109 (1990), 747–752. DOI 10.1090/S0002-9939-1990-1012927-4 | MR 1012927
[6] J. C.  Ferrando: When does $( \Sigma ,X)$ contain a copy of  $\ell _{\infty }$. Math. Scand. 74 (1994), 271–274. DOI 10.7146/math.scand.a-12496 | MR 1298368
[7] P. Habala, P.  Hájek, and V. Zizler: Introduction to Banach Space. Matfyzpress, Prague, 1996.
[8] E. Hewitt, K.  Stromberg: Real and Abstract Analysis. Graduate Texts in Mathematics  25. Springer-Verlag, New York-Heidelberg-Berlin, 1975. MR 0367121
[9] K.  Musial: The weak Radon-Nikodým property in Banach spaces. Stud. Math. 64 (1979), 151–173. DOI 10.4064/sm-64-2-151-174 | MR 0537118 | Zbl 0405.46015
[10] E. Saab, P.  Saab: On complemented copies of  $c_{0} $ in injective tensor products. Contemp. Math. 52 (1986), 131–135. DOI 10.1090/conm/052/840704
[11] M.  Talagrand: Quand l’espace des mesures a variation bornée est-it faiblement sequentiellement complet. Proc. Am. Math. Soc. 90 (1984), 285–288. (French) MR 0727251
Partner of
EuDML logo