Homological Methods in Banach Space Theory

Approaches Banach space theory using methods from homological algebra, with concrete examples and proofs of many new and classical results.

Félix Cabello Sánchez (Author), Jesús M. F. Castillo (Author)

9781108478588, Cambridge University Press

Hardback, published 26 January 2023

500 pages
23.5 x 15.8 x 3.9 cm, 0.98 kg

'This masterpiece seemingly is an excellent math book saturated with lush insights and marvellous perspectives on Banach spaces. But start to read it. When furniture and people start to appear to you as diagrams you will realize that you are in the middle of a fascinating wildlife event.' Piotr Koszmider, Institute of Mathematics of the Polish Academy of Sciences

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.

Preliminaries
1. Complemented subspaces of Banach spaces
2. The language of homology
3. Quasilinear maps
4. The functor Ext and the homology sequences
5. Local methods in the theory of twisted sums
6. Fraissé limits by the pound
7. Extensions of operators, isomorphisms and isometries
8. Extension of C(K)-valued operators
9. Singular exact sequences
10. Back to Banach space theory
Bibliography
Index.

Subject Areas: Numerical analysis [PBKS], Functional analysis & transforms [PBKF], Algebra [PBF]