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Lebesgue Spaces

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Book cover An Introductory Course in Lebesgue Spaces

Part of the book series: CMS Books in Mathematics ((CMSBM))

Abstract

Lebesgue spaces are without doubt the most important class of function spaces of measurable functions. In some sense they are the prototype of all such function spaces. In this chapter we will study these spaces and this study will be used in the subsequent chapters. After introducing the space as a normed space, we also obtain denseness results, embedding properties and study the Riesz representation theorem using two different proofs. Weak convergence, uniform convexity, and the continuity of the translation operator are also studied. We also deal with weighted Lebesgue spaces and Lebesgue spaces with the exponent between 0 and 1. We give alternative proofs for the Hölder inequality based on Minkowski inequality and also study the Markov, Chebyshev, and Minkowski integral inequality.

There is much modern work, in real or complex function theory, in the theory of Fourier series, or in the general theory of orthogonal developments, in which the ‘Lebesgue classes L k ’ occupy the central position. Godfrey Harold Hardy, John Edensor Littlewood & George Pólya

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Authors and Affiliations

  1. Universidad Nacional de Colombia, Bogotá, Colombia

    René Erlín Castillo

  2. Pontificia Universidad Javeriana, Bogotá, Colombia

    Humberto Rafeiro

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  1. René Erlín Castillo
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Castillo, R., Rafeiro, H. (2016). Lebesgue Spaces. In: An Introductory Course in Lebesgue Spaces. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-30034-4_3

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