Homological Methods in Banach Space Theory

This book provides functional analysts with a new perspective on their field and tools to tackle problems, introducing techniques and constructions from homological algebra and category theory. It presents classical results and advances, and applies them to solve old and new problems in (quasi-) Banach spaces, outlining new lines of research.

Format: Hardback
Length: 500 pages
Publication date: 26 January 2023
Publisher: Cambridge University Press

Many researchers in geometric functional analysis are unaware of the algebraic aspects of the subject and the remarkable advances they have permitted in the last half century. This book, written by two world experts on homological methods in Banach space theory, offers functional analysts a fresh perspective on their field and powerful tools to tackle its challenges. 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 applied to present both important classical results and groundbreaking advances from recent years. Finally, the authors use them to solve many old and new problems in the theory of (quasi-) Banach spaces and outline new lines of research. Containing a wealth 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.

Introduction:
Geometric functional analysis is a vibrant field that has made significant contributions to mathematics and its applications. However, there is a growing concern among researchers in this area that many are unaware of the algebraic aspects of the subject and the remarkable advances they have permitted in the last half century. This book aims to address this issue by providing functional analysts with a comprehensive introduction to homological algebra and its applications in Banach space theory.
What is Homological Algebra?:
Homological algebra is a branch of mathematics that deals with the study of algebraic structures, particularly those that arise in the context of differentiable manifolds and vector bundles. It is a powerful tool that allows functional analysts to study the properties and behavior of complex systems, such as operators, Banach spaces, and partial differential equations. Homological algebra provides a framework for analyzing and manipulating these structures using techniques such as homomorphisms, homology, and cohomology.
Applications in Banach Space Theory:
Banach space theory is a fundamental part of geometric functional analysis and has played a crucial role in the development of the field. Homological algebra provides a rich set of tools that can be used to study the properties and behavior of Banach spaces, including their normality, compactness, and the theory of operator algebras. By applying homological algebra to Banach space theory, functional analysts can obtain new insights into the behavior of operators and the solutions of partial differential equations.
Benefits of Learning Homological Algebra:
Learning homological algebra offers functional analysts a number of benefits. Firstly, it provides them with a new perspective on their field, allowing them to see problems from a different angle. Secondly, it offers them new tools to tackle complex problems, such as the analysis of operators and the solutions of partial differential equations. Thirdly, it enables them to work with a wide range of mathematical structures, including those that arise in other fields such as algebraic geometry and topology.
Conclusion:
In conclusion, this book provides functional analysts with a comprehensive introduction to homological algebra and its applications in Banach space theory. By learning homological algebra, functional analysts can gain a new perspective on their field, obtain new tools to tackle complex problems, and work with a wide range of mathematical structures. This book is an essential resource for anyone interested in geometric functional analysis and its applications.

Weight: 992g
Dimension: 236 x 159 x 39 (mm)
ISBN-13: 9781108478588