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On the Radon-Nikodym-property, and related topics in locally convex spaces

Part of the Lecture Notes in Mathematics book series (LNM,volume 645)

Abstract

We introduce L 1X (μ), the space of classes of X-valued μ-integrable functions used by Saab, which is an extension of the space of classes of Bochner-integrable functions, in Banach spaces. X denotes here a sequentially complete locally convex space.

We give examples of spaces which are dentable, σ-dentable, having the Radon-Nikodym-Property, or having the Bishop-Phelps-Property, by proving some projective limit results.

We also prove the following theorem : The following implications are valid :

$$(i) \Leftrightarrow (ii) \Rightarrow (iii) \Leftrightarrow (iv) \Leftrightarrow (v)$$
  1. (i)

    X has the Radon-Nikodym-Property.

  2. (ii)

    Every uniformly bounded martingale is L 1X -convergent.

  3. (iii)

    Every uniformly bounded martingale is L 1X -Cauchy.

  4. (iv)

    Every uniformly bounded and finitely generated martingale is L 1X -Cauchy.

  5. (v)

    X is σ-dentable.

So we have the equivalency of (i) through (v) for quasi-complete (BM)-spaces.

Keywords

  • Banach Space
  • Convex Space
  • Factor Space
  • Projective Limit
  • Finite Partition

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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© 1978 Springer-Verlag

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Egghe, L. (1978). On the Radon-Nikodym-property, and related topics in locally convex spaces. In: Aron, R.M., Dineen, S. (eds) Vector Space Measures and Applications II. Lecture Notes in Mathematics, vol 645. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069664

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  • DOI: https://doi.org/10.1007/BFb0069664

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08669-7

  • Online ISBN: 978-3-540-35903-6

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