Sánchez F.C., Castillo J.M.F. Homological Methods in Banach Space Theory

Cambridge University Press, 2023. — 559 p. — (Cambridge Studies in Advanced Mathematics 203). — ISBN 9781108478588.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.Preface
Preliminaries
Complemented Subspaces of Banach Spaces
The Language of Homology
Quasilinear Maps
The Functor Ext and the Homology Sequences
Local Methods in the Theory of Twisted Sums
Fraïsse Limits by the Pound
Extension of Operators, Isomorphisms and Isometries
Extension of C(K)-Valued Operators
Singular Exact Sequences
Back to Banach Space Theory