This is the first of a three volume introduction to analysis, which appeared recently in English translation, after 2 editions in German. It is a wonderful book that distinguishes itself by the clarity of presentation, by the fact that it is self-contained and has many exercises at various degrees of difficulty (among many other great qualities.)
The book is intended to be used both as a self study and as a textbook for a course in analysis. The consequence of this goal is the thorough treatment (with complete and elegant proofs) of many topics which do not usually appear in an analysis textbook.
Chapter I FOUNDATIONS is a thorough discussion about the number systems (from the natural numbers to the complex numbers), including also the fundamentals of set theory and the algebra needed. This is a very elegant and complete treatment of topics usually not found in analysis books. Also, it is a wonderful way of emphasizing the connections between logic, algebra, analysis and a great introduction for the chapters to come. The exercises are plenty and very well selected.
The chapters to follow are "regular" chapters expected in an analysis text&but many times with more topics than usually found in such books. For instance, Chapter II CONVERGENCE includes the expected convergence of sequences and series sections, but also a section called "Normed Vector Spaces" which ends by discussing "Convergence in Product Spaces". Another example can be found in Chapter III CONTINUOUS FUNCTIONS, which is much longer than in most texts because it includes sections like "The Fundamentals of Topology", "Compactness", "Connectivity", all treated very rigorously, elegantly, with many applications and well-chosen exercises. Chapter V SEQUENCES OF FUNCTIONS introduces such concepts as Banach Spaces and Banach Algebras, not very often discussed in Analysis I.
In summary, we have here a very-very good, self-contained, well written book, including many applications and exercises, that can be used as a text for courses (real analysis, foundations, etc), as self-study, and/or as a basis for research in mathematics or even other disciplines. This book is a wonderful read.
Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu