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Banach Space Theory pp 179–235Cite as

Schauder Bases

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Abstract

In this chapter we shall introduce Schauder bases, an important concept in Banach space theory. Elements of a Banach space with a Schauder basis may be represented as infinite sequences of “coordinates,” which is very natural and useful for analytical work. Although not every separable Banach space admits a Schauder basis (this is a deep and difficult result of Enflo), basic sequences exist in every infinite-dimensional Banach space (Mazur) and are ideal for the study of linear subspaces and quotients. In many respects, the assumption of having a Schauder basis is not very restrictive, and in fact, for naturally defined separable Banach spaces, it is usually easy to find their Schauder basis. This notion has proved to be an extremely useful tool in the study of the structure of classical as well as abstract Banach spaces.

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Authors and Affiliations

  1. Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, Praha 1, 11567, Prague, Czech Republic

    Marián Fabian & Petr Hájek

  2. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 16627, Prague, Czech Republic

    Petr Habala

  3. Departamento de Matematica Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    Vicente Montesinos

  4. Department of Mathematical and Statistical Sciences, University of Alberta, Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Václav Zizler

Authors
  1. Marián Fabian
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  2. Petr Habala
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  3. Petr Hájek
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  4. Vicente Montesinos
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  5. Václav Zizler
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Correspondence to Marián Fabian .

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Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V. (2011). Schauder Bases. In: Banach Space Theory. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7515-7_4

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