This is the second of a there volume introduction to analysis, which appeared recently in English translation, after two 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 first volume appeared in English translation in 2005.
The book is intended to be used both as a self study and as a textbook for courses in analysis. The consequence of this goal is the thorough treatment (with complete and elegant proofs) of many topics from analysis.
The chapters in this second volume follow logically the ones from the previous volume. Even the numbering of the chapters is in continuation:
Chapter VI: Integral Calculus in One Variable
Chapter VII: Multivariable Differential Calculus
Chapter VIII: Line Integrals
There are only three chapters in this book, but the depth of the treatment, the richness of examples and exercises, the rigor of the proofs, make it too long for a one-semester course. Instructors could choose parts of the book to be used in several courses, or (better yet) this book could be used for independent studies, or self-studies by students, or even by more experienced mathematicians in their study and research.
Every student of analysis could benefit from reading (parts of) this book, and every analyst will find useful to own this book, as a reference when teaching an analysis course, or as a source of information.
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 associate 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.