Skip to main content
SpringerLink
Log in

Liapounoff Convexity-type Theorems

  • Conference paper
Book cover Vector Measures, Integration and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 201))

Abstract

We investigate Liapounoff convexity-type theorems for the case of Fréchet space-valued measures defined on Fields of sets. We extend theorems by I. Kluvánek [10], J.J. Uhl [21] and S. Ohba [15] to this case. Results relating bounded finitely additive measures with strongly additive measures is also of interest.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Utpal K. Bandyopadhyay. On vector measures with the Darboux property. Quart. J. Math. Oxford Ser. (2), 25:57–60, 1974.

    Google Scholar 

  2. K.P.S. Bhaskara Rao and M. Bhaskara Rao. Theory of charges, volume 109 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983.

    Google Scholar 

  3. J. Diestel. Applications of weak compactness and bases to vector measures and vectorial integration. Rev. Roumaine Math. Pures Appl., 18:211–224, 1973.

    MATH  MathSciNet  Google Scholar 

  4. J. Diestel and J.J. Uhl, Jr. Vector measures. American Mathematical Society, Providence, R.I., 1977.

    MATH  Google Scholar 

  5. N. Dinculeanu. Vector measures. International Series of Monographs in Pure and Applied Mathematics, Vol. 95. Pergamon Press, Oxford, 1967.

    Google Scholar 

  6. Nelson Dunford and Jacob T. Schwartz. Linear Operators. I. General Theory. With the assistance of W.G. Bade and R.G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York, 1958.

    MATH  Google Scholar 

  7. A. Fernández and F. Naranjo. Rybakov’s theorem for vector measures in Fréchet spaces. Indag. Math. (N.S.), 8(1):33–42, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  8. Igor Kluvánek. The extension and closure of vector measure. In Vector-and operator-valued measures and applications (Proc. Sympos., Alta, Utah, 1972), pages 175–190. Academic Press, New York, 1973.

    Google Scholar 

  9. Igor Kluvánek. The range of a vector-valued measure. Math. Systems Theory, 7:44–54, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  10. Igor Kluvánek. The range of a vector measure. Bull. Amer. Math. Soc., 81:609–611, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  11. Igor Kluvánek and Greg Knowles. Vector measures and control systems. North-Holland Publishing Co., Amsterdam, 1976. North-Holland Mathematics Studies, Vol. 20, Notas de Matemática, No. 58. [Notes on Mathematics, No. 58].

    MATH  Google Scholar 

  12. A. Liapounoff. Sur les fonctions-vecteurs complètement additives. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR], 4:465–478, 1940.

    MathSciNet  Google Scholar 

  13. Joram Lindenstrauss. A short proof of Liapounoff’s convexity theorem. J. Math. Mech., 15:971–972, 1966.

    MATH  MathSciNet  Google Scholar 

  14. S. Mazur and W. Orlicz. Sur les espaces métriques linéaires. II. Studia Math., 13:137–179, 1953.

    MATH  MathSciNet  Google Scholar 

  15. Sachio Ohba. The range of vector measure and Radon-Nikodým property. Rep. Fac. Engrg. Kanagawa Univ., (16):4–5, 1978.

    MATH  MathSciNet  Google Scholar 

  16. W.J. Ricker. Rybakov’s theorem in Fréchet spaces and completeness of L 1-spaces. J. Austral. Math. Soc. Ser. A, 64(2):247–252, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  17. Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York, second edition, 1991.

    MATH  Google Scholar 

  18. E. Saab and P. Saab. Lyapunov convexity type theorems for non-atomic vector measures. Quaest. Math., 26(3):371–383, 2003.

    MATH  MathSciNet  Google Scholar 

  19. G.L. Seever. Measures on F-spaces. Trans. Amer. Math. Soc., 133:267–280, 1968.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Sobczyk and P.C. Hammer. A decomposition of additive set functions. Duke Math. J., 11:839–846, 1944.

    Article  MATH  MathSciNet  Google Scholar 

  21. J.J. Uhl, Jr. The range of a vector-valued measure. Proc. Amer. Math. Soc., 23:158–163, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  22. Manuel Valdivia. Topics in locally convex spaces, volume 67 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1982. Notas de Matemática [Mathematical Notes], 85.

    Google Scholar 

  23. P. Wojtaszczyk. Banach spaces for analysts, volume 25 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, 0002, South Africa

    Rudolf G. Venter

Authors
  1. Rudolf G. Venter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Facultad de Matemáticas, Universidad de Sevilla, 41080, Sevilla, Spain

    Guillermo P. Curbera

  2. Fachbereich 6: Mathematik, Universität Siegen, 57068, Siegen, Germany

    Gerd Mockenhaupt

  3. Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85072, Eichstätt, Germany

    Werner J. Ricker

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Venter, R.G. (2009). Liapounoff Convexity-type Theorems. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol 201. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0211-2_34

Download citation

Publish with us

Policies and ethics

Search

Navigation