Skip to main content
SpringerLink
Log in

Norm-attaining functionals onC(Q, X)

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

Functionals (vector measures) defined on the spaceC(Q, X) of continuous abstract functions (whereQ is a compact Hausdorff space andX is a Banach space) and attaining their norm on the unit sphere are considered. A characterization of such functionals is given in terms of the Radon-Nikodym derivative of the vector measure with respect to the variation of the measure and in terms of analogs of the derivative. Applications to the characterization of finite-codimensional subspaces with the best approximation property are given. Similar results are obtained for the spaceB(Q, Σ, X) of uniform limits of simple functions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Diestel,Geometry of Banach Spaces, Spinger-Verlag, Berlin-New York (1975).

    Google Scholar 

  2. I. Singer,Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, Berlin (1970).

    Google Scholar 

  3. G. M. Ustinov and Yu. A. Shashkin, “Norm-attaining functionals,” in:Studies in Functional Analysis and Applications, Collection of Scientific Papers [in Russian], Ural State University, Sverdlovsk (1985), pp. 103–109.

    Google Scholar 

  4. G. M. Ustinov, “Existence subspaces inC(Q, X),” in:Studies in Functional Analysis and Topology, Collection of Scientific Papers [in Russian], Ural State University, Sverdlovsk (1990), pp. 127–130.

    Google Scholar 

  5. L. P. Vlasov, “The existence of best approximation elements inC(Q, X),”Mat. Zametki [Math. Notes],58, No. 2, 163–175 (1995).

    MATH  MathSciNet  Google Scholar 

  6. I. Singer, “Linear functionals on the space of continuous mappings of a compact Hausdorff space into a Banach space,”Revue Roun. Math. Pur. Appl.,2, 301–315 (1957).

    MATH  Google Scholar 

  7. N. Bourbaki,Integration (Vector Integration. Haar Measure. Convolution and Representations) [Russian translation], Nauka, Moscow (1970).

    Google Scholar 

  8. N. Dunford and J. Schwartz,Linear Operators. General Theory, New York (1962).

    Google Scholar 

  9. V. S. Balaganskii and L. P. Vlasov,Approximation-Geometric Properties of Sets in Banach Spaces [in Russian], Preprint, Institute for Mathematics and Mechanics, Ural Division of the USSR Academy of Sciences, Sverdlovsk (1990).

    Google Scholar 

  10. C. Bessaga and A. Pelczyński, “Selected topics in infinite-dimensional topology,” in:Monogr. Mat., Vol. 58, PWN, Warsawa (1975).

    Google Scholar 

  11. R. Engelking,General Topology, PWN, Warszawa (1983).

    Google Scholar 

  12. A. V. Arkhangel'skii and V. I. Ponomarev,Foundations of General Topology in Problems and Exercises [in Russian], Nauka, Moscow (1974).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences, USSR

    L. P. Vlasov

Authors
  1. L. P. Vlasov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 45–56, January, 1997.

Translated by V. E. Nazaikinskii

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vlasov, L.P. Norm-attaining functionals onC(Q, X) . Math Notes 61, 38–47 (1997). https://doi.org/10.1007/BF02355006

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02355006

Key words

Search

Navigation