This book is unlike any other--as the author says, it's much like a collection of first chapters of graduate-level math texts. What makes this book unique and valuable are the connections drawn between all the different areas, as well as elementary presentations of certain key topics. I feel that this book, much like Lang's algebra, has the potential to redefine the way analysis is studied and taught.
Most analysis books go into much more depth and prove many more results--this book, in spite of its size, won't take you very far in the field, but the knowledge it cultivates is deep, and it will give you both breadth and a modern perspective. Instead of accepting one set of axioms, or one set of structures, and then proving a bunch of results, this book explores many different possible combinations of axioms, and structures, and explores which results hold in which cases. Numerous examples of some of the more abstract mathematical objects are provided. The book shows how different branches of mathematics relate to each other, drawing together set theory, algebra, topology, category theory, and many of the major branches of analysis.
Some parts of the book are elementary, focusing on the very foundations of mathematical ideas, language, notation, and proof, and these parts will be indispensable to young, maturing mathematicians. The discussion of constructivism and intangible objects is indispensable and is among the best exposition of these topics I have found in any texts; many functional analysis books make frequent use of intangibles without explicitly discussing them.
Other parts of this book are advanced and will be useful to experts. The chapter on convergence is an absolute gem, thoroughly exploring the relationship between the net and filter views of convergence. The material on measures and integration is highly general, probably more useful for experts than for beginning students. Overall, the writing is exceptionally clear: the material is presented from an introductory level, but the author presents it exhaustively, not glossing over exceptions or special cases. This book is simply unparalleled in the way it presents so much abstract material and yet is so easy to read.
Possibly the best aspect of this book is that it is up-to-date. Many analysis books have changed little since the 1950's. However, the field of analysis has changed greatly. If you're unfortunate enough to be stuck in a class or a program in which you are being taught the "old-fashioned way", this book will help you get up to speed with some of the more modern ways of looking at things. While it is not comprehensive (it barely touches topics like classical complex analysis or probability, and it will probably be less useful for applied mathematicians and people in related disciplines), I still think it is the single most important math book on my shelf. I would recommend every serious mathematician to purchase this book.