On copies of \( c_0 \) in some function spaces

Article

Abstract

If (Ω, Σ, µ) is a probability space and X a Banach space, a theorem concerning sequences of X-valued random elements which do not converge to zero is applied to show from a common point of view that the F-normed space \( L_p (\mu ,X) \) of all classes of X-valued random variables, as well as the p-normed space Lp(µ, X) of all X-valued p-integrable random variables with 0 < p < 1 and the space \( P_1 (\mu ,X) \) of the µ-measurable X-valued Pettis integrable functions, all contain a copy of \( c_0 \) if and only if X does. We also show that if is a noncompact hemicompact topological space, then the Banach space \( C_0 (\Omega ) \) of all scalarly valued continuous functions defined on vanishing at infinity, equipped with the supremum-norm, contains a norm-one complemented copy of \( c_0 \) .

Keywords

Banach space copy of c0 vector-valued random element F-norm p-norm space Lp(µ, X) with 0≤ p<1 space P1(µ, Xhemicompact space space C0(Ω) 

Mathematics Subject Classifications

46E40 46B09 

Sobre copias de \( c_0 \) en algunos espacios de funciones

Resumen

Si (Ω, Σ, µ) es un espacio de probabilidad y X un espacio de Banach, aplicamos un teorema sobre sucesiones de variables aleatorias con valores en X que no convergen a cero para demostrar, desde un punto de vista común, que el espacio F-normado \( L_p (\mu ,X) \) de las variables aleatorias p-integrables X-valoradas, con 0 < p < 1, y el espacio \( P_1 (\mu ,X) \) de las funciones X-valoradas µ-medibles y Pettis integrables, todos ellos contienen una copia de \( c_0 \) si y sólo si X también la contiene. Asimismo probamos que si Ω es un espacio topológico hemicompacto no compacto, el espacio de Banach \( C_0 (\Omega ) \) de las funciones continuas con valores escalares definidas en Ω que se anulan en el infinito, equipado con la norma supremo, contiene una copia de c0 norma-uno complementada.

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References

  1. [1]
    Diestel, J., (1984). Sequences and series in Banach spaces, Springer-Verlag. GTM 92 New York Berlin Heidelberg Tokyo.Google Scholar
  2. [2]
    Drewnowski, L. and Labuda, I., (2002). Topological vector spaces of Bochner measurable functions, Illinois J. Math, 46, 287–318.MATHMathSciNetGoogle Scholar
  3. [3]
    Dunford, N. and Schwartz, J. T., (1988). Linear Operators, I. Wiley-Interscience.Google Scholar
  4. [4]
    Ferrando, J. C., (2002). On sums of Pettis integrable random elements, Quaestiones Math., 25, 311–316.MATHMathSciNetGoogle Scholar
  5. [5]
    Ferrando, J. C. and López Pellicer, M., (2001). Complemented copies of C0 in C0 (Ω), Rev. R. Acad Cien. Serie A Mat., 95, 13–17.MATHGoogle Scholar
  6. [6]
    Freniche, F. J., (1998). Embedding c0 in the space of Pettis integrable functions, Quaestiones Math., 21, 261–267.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Hoffmann-Jørgensen, J., (1974). Sums of independent Banach space valued random variables, Studia Math., 52, 159–186.MathSciNetGoogle Scholar
  8. [8]
    Köthe, G., (1983). Topological Vector Spaces I, Grundlehren der mathematischen Wissenschaften, 159. Springer-Verlag. Berlin Heidelberg New York.Google Scholar
  9. [9]
    Kwapień, S., (1974). On Banach spaces containing c0, Studia Math., 52, 187–188.MATHMathSciNetGoogle Scholar
  10. [10]
    Labuda, I., (1979). Spaces of measurable functions, Comment. Math., Special Issue 2, 217–249.MathSciNetGoogle Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.Centro de Investigación OperativaUniversidad Miguel HernándezElche (Alicante)Spain

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