'This is a remarkable book whose concept is not to present a comprehensive overview of a field but, keeping the geographical metaphor, it follows one of the main rivers there, exploring its tributaries and distributaries and going even beyond the boundaries of the original territory. In this case, the author follows the river of completeness of normed spaces and presents its sources in more fundamental areas of mathematics and its impact on an array of theoretical and applied fields. The book is written in an effortless, casual, yet rigorous style and is a pleasure to read.' Jacek Banasiak, University of Pretoria 'This non-standard book can serve as a supplement to an introductory functional analysis course. Centred around the fundamental notion of completeness, it introduces a variety of basic notions and tools, and covers a solid range of material presented in a lively and vivid way. The emphasis has been placed on applications of abstract concepts, including the rudiments of operator semigroups, partial differential equations, and probability theory. The exposition is friendly and fresh, and many insightful comments facilitate digesting the material. Each chapter is accompanied by helpful summaries and carefully selected exercises. The book would be useful for beginners aiming to improve their understanding of the subject, for lecturers willing to refresh their routine courses, and it would be of value to anyone who likes elegant expositions of important mathematical theories.' Yuri Tomilov, Institute of Mathematics, Polish Academy of Sciences
