Foundations of Real and Abstract Analysis

While at times the logic of this book is very clear, I would not say this is a good book. I would not recommend it for self study, and I think it would be a terrible choice of a textbook at any level. It might be useful for someone who already knows the material and wants a concise review of it, but I do not think it would make a useful reference either. The philosophy of the author is downright offensive. The author writes with an inflated view of his own areas of expertise; in addition to including a few explicit rants about the lack of students' background and interest in real and functional analysis, this attitude shows through in his presentation of much of the material without motivation or discussion. There is next to no mention of any of the rich connections of the subjects covered to other branches of mathematics. The author holds the annoying (and I believe very wrong) notion that mathematical analysis is a field of study in which intuition is inherently flawed and should be made subordinate to logic. Instead of helping the reader develop proper intuition, the book focuses exclusively on logic. The book follows a dry theorem-proof based approach, is almost completely linear, and the proofs emphasize concise logical presentation at the expense of understanding the motivation for and underlying meaning of the results. The one-chapter review of basic analysis is not very useful--any student already knowing the material will not need it, but those learning it for the first time would find it difficult, uninteresting, and non-comprehensive. However, I also find later chapters disappointing. Measure theory is introduced slightly differently from many books, but there is no discussion of why the particular path was chosen. Many of the usual proofs that have a clear intuitive interpretation are presented without that interpretation--the reader is left with the ability to follow or memorize the proofs, but without the ability to understand them or construct them on her own. Later chapters are made to depend on earlier ones in a most rigid fashion, making this book difficult to use as a reference. The book presents one particular view of the material without preparing the reader for encountering other perspectives. In short, there is nothing fresh and new that this book has to offer, and it is deficient in many ways. If you are a crotchety old man who likes to rant about the low quality of mathematics students these days, you might just like this book. Otherwise, just pick up one of the many excellent analysis texts out there. Rudin's has much more motivation (and imagination!) and focuses more on intuition than this text, and it somehow does this while covering more material. Royden's book is the classic--it has much more explanation and is very easy to read. One might also look at one of the texts that emphasise applications of measure theory to probability, or one of the many texts that develops measure theory in the context of functional analysis.