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Integral Representation of Linear Functionals

  • Boris Makarov
  • Anatolii Podkorytov
Chapter
  • 4.3k Downloads
Part of the Universitext book series (UTX)

Abstract

This chapter is devoted to questions relevant to both measure theory and functional analysis. In Sect. 12.1, we develop a general scheme allowing one to obtain integral representations for order continuous functionals on functional spaces, in particular, in the spaces \(\mathcal{L}^{p}\) for finite p. In Sect. 12.2, we prove that a positive functional on the space of continuous functions defined on a locally compact space has an integral representation. In Sect. 12.3, we describe the general form of continuous linear functionals in the spaces of functions continuous on a compact spaces and also in the spaces \(\mathcal{L}^{p}\) for 1⩽p<∞. As a consequence, we prove that the Borel charges on a multi-dimensional torus are determined by their Fourier coefficients. We consider various applications of these results to harmonic analysis.

Keywords

Measurable Function Integral Representation Real Function Dual Space Compact Space 
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.

References

  1. [B-I]
    Borisovich, Yu.G., Bliznyakov, N.M., Fomenko, T.N., Izrailevich, Ya.A.: Introduction to Topology. Mir, Moscow (1985) Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Boris Makarov
    • 1
  • Anatolii Podkorytov
    • 1
  1. 1.Mathematics and Mechanics FacultySt Petersburg State UniversitySt PetersburgRussia

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