This book provides an elementary introduction to both linear and nonlinear functional analysis. The authors emphasize the variability of infinite dimensional spaces and highlight the role of compactness. The linear part of the book covers classical means of doing functional analysis (normed linear spaces and operators, the criteria of compactness in varied spaces and the Riesz-Schauder theory of compact operators). The second part of the book concentrates mainly on fixed point theorems (Banach, Brouwer, Schauder, Darbo, Borsuk). The appendix includes the Baire category theorem, bases in Banach spaces, the Weierstrass approximation theorem and the Tietze-Urysohn and Dugundji extension theorems.

Each chapter ends with many exercises and problems of varying difficulty, which give further applications and extensions of the theory. The book is easily understandable and it can be warmly recommended to graduate students in mathematics, physics, biology, chemistry and engineering as a neat introduction to functional analysis.

Reviewer:

jl