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2 results

  • 2 - The Language of Homology

  • 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 46-127

  • Summary

    The chapter introduces the basic elements of the homological language and translates the statements about complemented and uncomplemented subspaces presented in Chapter 1 into this language. The reader will find everything they need to know at this stage about exact sequences, categorical and homological constructions for absolute beginners and injective and projective Banach and quasi-Banach spaces. The chapter describes the basic homological constructions appearing in nature: complex interpolation, the Nakamura-Kakutani, Foia\c{s}-Singer, Pe\l czy\’nski-Lusky and Bourgain’s $\ell_1$ sequences, the Ciesielski-Pol, Bell-Marciszewski and Bourgain-Pisier constructions, the Johnson-Lindenstrauss spaces and so on. A good number of advanced topics are included: diagonal and parallel principles for exact sequences, the Device, 3-space results, extension and lifting of operators, $M$-ideals and vector-valued Sobczyk’s theorems

Homological Methods in Banach Space Theory
  • Félix Cabello Sánchez, Jesús M. F. Castillo
  • Published online:
    19 January 2023
    Print publication:
    26 January 2023
  • Many researchers in geometric functional analysis are unaware of algebraic aspects of the subject and the advances they have permitted in the last half century. This book, written by two world experts on homological methods in Banach space theory, gives functional analysts a new perspective on their field and new tools to tackle its problems. All techniques and constructions from homological algebra and category theory are introduced from scratch and illustrated with concrete examples at varying levels of sophistication. These techniques are then used to present both important classical results and powerful advances from recent years. Finally, the authors apply them to solve many old and new problems in the theory of (quasi-) Banach spaces and outline new lines of research. Containing a lot of material unavailable elsewhere in the literature, this book is the definitive resource for functional analysts who want to know what homological algebra can do for them.