Integral Representation of Linear Functionals

  • Boris Makarov
  • Anatolii Podkorytov
Part of the Universitext book series (UTX)


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.


Measurable Function Integral Representation Real Function Dual Space Compact Space 
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  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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