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  • 1 - Complemented Subspaces of Banach Spaces

  • Félix Cabello Sánchez, Universidad de Extremadura, Spain, Jesús M. F. Castillo, Universidad de Extremadura, Spain
  • Book:
    Homological Methods in Banach Space Theory
    Published online:
    19 January 2023
    Print publication:
    26 January 2023, pp 9-45

  • Summary

    The chapter contains the fundamental results about Banach and quasi-Banach spaces and their complemented subspaces that are necessary for this book. Classical topics included are the Aoki-Rolewicz theorem, the completion of a quasinormed space, $p$-Banach envelopes, Pe\l czy\’nski’s decomposition method, uncomplemented subspaces of classical spaces, indecomposable spaces, type and cotype of quasi-Banach spaces, local properties, ultraproducts, the Dunford-Pettis and Grothendieck properties, properties (V) of Pe\l czy\’nski and Rosenthal, $C(K)$-spaces and their complemented subspaces and so on. More advanced topics have been also included, such as Sobczyk’s theorem and its non-separable derivatives and ultrapowers, mainly of the $L_p$-spaces.