The book under review is truly marvelous. Captivatingly engaging, it weaves an interesting, lively, and crystal clear sequence of ideas comprising the heart of modern analysis. The order of presentation is so carefully chosen and the exposition is so masterful as to possess the traits of a literary art form. The new ideas in each chapter seem like an inevitability, following up precisely where the previous chapter ended. Coupled with hundreds of exercises to hone and delight the reader this book is highly recommended reading, offering countless hours of fun for the serious student seeking to master the elements of modern analysis.
The book can certainly be said to be very modest in its prerequisites. In fact, many fundamental notions of calculus and linear algebra are included in the text, so the treatment is quite self-contained. Some mathematical maturity is certainly desirable in order to more efficiently swim through the ocean of results rather than drown in details, but the first few chapters, which are more elementary and rather slow paced, will serve to get any reader quickly on track for the rest of the book.
The topics covered are varied, including measure theory, topology, metric spaces, normed spaces, Hilbert and Banach spaces, probability theory, harmonic analysis, wavelet theory, information theory, and then some. The progression is from the concrete to the abstract, constantly building on the previous results to motivate and illustrate the presentation of new concepts.
The book has many more points in its favor. Each chapter begins with a biography of a notable mathematician whose contributions are relevant to the chapter. The choice of topics is fine-tuned to suit the student and prepare her for taking the first steps in the forefront of analysis research. Each section of each chapter has a clear objective, presenting a highly balanced amount of new content, while constantly revisiting previous ideas and results. And again, in my view what makes the reading of this book a truly invaluable and enjoyable experience for students is the masterful weaving of a tremendous amount of intricate mathematics into a coherent linear progression spanning over 600 pages, in such a way that at every step the new results and concepts fall so neatly into place that the reader will have to be reminded that the historical development of these ideas was much more painful than the book would make one believe.
The book is highly suited for self-study as well as a book complementing either introductory courses in modern analysis or more advanced such courses. The book can also very effectively be used as a main text book for a sequence of courses in analysis. While the book is aimed at students, who will certainly benefit most from it, more proficient mathematicians are likely to find the book very enjoyable as well.
Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.