## Embedding $$c_{0}$$ in $$\operatorname {bvca}(\Sigma ,X)$$.(English)Zbl 1174.46016

Summary: 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 $$\operatorname {bvca}( \Sigma ,X)$$, 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.

### MSC:

 46E27 Spaces of measures 46B25 Classical Banach spaces in the general theory
Full Text:

### References:

 [1] J. Bourgain: An averaging result for c0-sequences. Bull. Soc. Math. Belg., Sér. B 30 (1978), 83–87. · Zbl 0417.46019 [2] P. Cembranos, J. Mendoza: Banach Spaces of Vector-Valued Functions. Lecture Notes in Mathematics Vol. 1676. Springer-Verlag, Berlin, 1997. · Zbl 0902.46017 [3] J. Diestel: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, 92. Springer-Verlag, New York-Heidelberg-Berlin, 1984. · Zbl 0542.46007 [4] J. Diestel, J. Uhl: Vector Measures. Mathematical Surveys, No 15. Am. Math. Soc., Providence, 1977. · Zbl 0369.46039 [5] L. Drewnowski: When does ca({$$\Sigma$$}, Y ) contain a copy of or c0? Proc. Am. Math. Soc. 109 (1990), 747–752. · Zbl 0724.46041 [6] J. C. Ferrando: When does bvca({$$\Sigma$$}, X) contain a copy of Math. Scand. 74 (1994), 271–274. · Zbl 0828.46018 [7] P. Habala, P. Hájek, and V. Zizler: Introduction to Banach Space. Matfyzpress, Prague, 1996. · Zbl 0904.46001 [8] E. Hewitt, K. Stromberg: Real and Abstract Analysis. Graduate Texts in Mathematics 25. Springer-Verlag, New York-Heidelberg-Berlin, 1975. [9] K. Musial: The weak Radon-Nikodým property in Banach spaces. Stud. Math. 64 (1979), 151–173. [10] E. Saab, P. Saab: On complemented copies of c0 in injective tensor products. Contemp. Math. 52 (1986), 131–135. [11] M. Talagrand: Quand l’espace des mesures a variation bornée est-it faiblement sequentiellement complet? Proc. Am. Math. Soc. 90 (1984), 285–288. (In French.) · Zbl 0534.46017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.