Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EQUIVALENT NORMS IN A BANACH FUNCTION SPACE AND THE SUBSEQUENCE PROPERTY

  • Calabuig, Jose M. (Instituto Universitario de Matematica Pura y Aplicada Universitat Politecnica de Valencia) ;
  • Fernandez-Unzueta, Maite (Centro de Investigacion en Matematicas, A.C.) ;
  • Galaz-Fontes, Fernando (Centro de Investigacion en Matematicas, A.C.) ;
  • Sanchez-Perez, Enrique A. (Instituto Universitario de Matematica Pura y Aplicada Universitat Politecnica de Valencia)
  • Received : 2018.10.13
  • Accepted : 2019.05.31
  • Published : 2019.09.01
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Abstract

Consider a finite measure space (${\Omega}$, ${\Sigma}$, ${\mu}$) and a Banach space $X({\mu})$ consisting of (equivalence classes of) real measurable functions defined on ${\Omega}$ such that $f{\chi}_A{\in}X({\mu})$ and ${\parallel}f{\chi}_A{\parallel}{\leq}{\parallel}f{\parallel}$, ${\forall}f{\in}({\mu})$, $A{\in}{\Sigma}$. We prove that if it satisfies the subsequence property, then it is an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.

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References

  1. M. Ciesielski, P. Kolwicz, and A. Panfil, Local monotonicity structure of symmetric spaces with applications, J. Math. Anal. Appl. 409(2) (2014), no.2, 649-662. https://doi.org/10.1016/j.jmaa.2013.07.028
  2. Y. Deng, M. O'Brien, and V. G. Troitsky, Unbounded norm convergence in Banach lattices, Positivity 21 (2017), no. 3, 963-974. https://doi.org/10.1007/s11117-016-0446-9
  3. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511526138
  4. P. Foralewski, H. Hudzik, W. Kowalewski, and M. Wisla, Monotonicity properties of Banach lattices and their applications - a survey, in Ordered structures and applications, 203-232, Trends Math, Birkhauser/Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-27842-1_14
  5. D. H. Fremlin, Topological Riesz Spaces and Measure Theory, Cambridge University Press, London, 1974.
  6. H. Hudzik and R. Kaczmarek, Monotonicity characteristic of Kothe-Bochner spaces, J. Math. Anal. Appl. 349 (2009), no. 2, 459-468. https://doi.org/10.1016/j.jmaa.2008.08.053
  7. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin, 1973.
  8. W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces. Vol. I, North-Holland Publishing Co., Amsterdam, 1971.
  9. W. Rudin, Functional Analysis, second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
  10. A. C. Zaanen, Integration, completely revised edition of An introduction to the theory of integration, North-Holland Publishing Co., Amsterdam, 1967.
  11. A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, Berlin, 1997. https://doi.org/10.1007/978-3-642-60637-3