Named for Banach, who was one of the great mathematicians of the twentieth century, the concept of Banach spaces figures prominently in the study of functional analysis, having applications to integral and differential equations, approximation theory, harmonic analysis, convex geometry, numerical mathematics, analytic complexity, and probability theory.

Written by a distinguished specialist in functional analysis, this book presents a comprehensive treatment of the history of Banach spaces and (abstract bounded) linear operators. While other historical texts on the subject focus on developments before 1950, this one is mainly devoted to the second half of the 20^{th} century.

Banach space theory is presented as a part of a broad mathematics context, using tools from such areas as set theory, topology, algebra, combinatorics, probability theory, logic, etc. Equal emphasis is given to both spaces and operators. Numerous examples and counterexamples elucidate the scope of the underlying concepts. As a stimulus for further research, the text also contains many problems which have not been previously solved.

The book may serve as a reference for researchers and as an introduction for graduate students who want to learn Banach space theory with some historical flavor. Helpful information is provided for professors in preparing their own lectures on functional analysis.