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Rating: 
Summary: Could have been great
Review: I speak as a graduate student in applied math. I really like this book but was bothered by its flaws. Nevertheless, with a good instructor, this text can make for a good learning experience.Positives: The book is well organized. It builds in a reasonable way so that I could focus on the material in the book and develop my understanding as I went. The book is reasonably well contained. Outside of a reasonable level of basics (a BA or BS in math) the proofs and most of the problems use material developed earlier in the text. I found the book very interesting -- I especially liked the topics presented in the last few chapters. Negatives: Lots of typos - the author's errata sheet is woefully incomplete. Too few expamples. Too condensed - sometimes to the point of incomprehensibility or even error. The contents of a whole course may be condensed in to a single chapter or even a single section. Things to be aware of: You should be comfortable with advanced calculus, topology, set theory, and algebra (linear and modern). It also helps to have had some basic real analysis. I highly recommend that you've seen Fourier transforms, Dirac deltas (distributions), and continuous probability. You aren't going to learn these here - you're going to see how measure theory is applied to them.
Rating:
Summary: Could have been great
Review: I speak as a graduate student in applied math. I really like this book but was bothered by its flaws. Nevertheless, with a good instructor, this text can make for a good learning experience. Positives: The book is well organized. It builds in a reasonable way so that I could focus on the material in the book and develop my understanding as I went. The book is reasonably well contained. Outside of a reasonable level of basics (a BA or BS in math) the proofs and most of the problems use material developed earlier in the text. I found the book very interesting -- I especially liked the topics presented in the last few chapters. Negatives: Lots of typos - the author's errata sheet is woefully incomplete. Too few expamples. Too condensed - sometimes to the point of incomprehensibility or even error. The contents of a whole course may be condensed in to a single chapter or even a single section. Things to be aware of: You should be comfortable with advanced calculus, topology, set theory, and algebra (linear and modern). It also helps to have had some basic real analysis. I highly recommend that you've seen Fourier transforms, Dirac deltas (distributions), and continuous probability. You aren't going to learn these here - you're going to see how measure theory is applied to them.
Rating: 
Summary: Frustrating
Review: It seems that the higher up you go in the Mathematics curriculum, the poorer the books you meet. In my honest opinion, a book should help you learn and understand the material as quickly as possible. Otherwise, you might as well be given a list of definitions, stuck in a closed padded room and asked to come up with all the theorems by yourself. Unfortunately, there are too many graduate textbooks out there written by individuals who seem to have no desire to make the ideas they are trying to present as clear as possible. There's no educational philosophy. This book falls under that category. For example, this book is almost completely devoid of any examples. I don't know about you, but from example, is how I learn. I could go through this book much faster, if there were some decent examples. You can tell me a thousand times what a sigma algebra is, but if you don't give me some decent, worked-out examples which might tell me why tell me why I should learn it (other than because I'll fail the course), I'm going to forget the definition after 5 minutes. Secondly, it would help if there were more pictures. A picture is worth a thousand words. Third, some of the definitions are not worded as well as they should be. last night I spent ten minutes trying to figure out whether the definition for x-section Ex = {y in Y : (x,y) in E} meant that "for all x," or just "for some x?" It turned out it meant "for fixed x." But nowhere was that little tidbit of information written. Ten minutes may not sound like much, but if you have to read 10 pages before you get to pleasure of spending 10 hours with the homework problems, that translates into a lot of time you could spend doing other things if only this book were presented in a manner which would enable you to learn the material more efficiently. I give it two stars primarily because some of the homework problems aren't too bad. If you have a choice, have a look at Kolmogorov and Fomins book on Real analysis. It's not perfect, but the material in it is organized better. (It's not as DENSE) Plus it's a Dover book, and therefore much cheaper.
Rating:
Summary: Frustrating
Review: It seems that the higher up you go in the Mathematics curriculum, the poorer the books you meet. In my honest opinion, a book should help you learn and understand the material as quickly as possible. Otherwise, you might as well be given a list of definitions, stuck in a closed padded room and asked to come up with all the theorems by yourself. Unfortunately, there are too many graduate textbooks out there written by individuals who seem to have no desire to make the ideas they are trying to present as clear as possible. There's no educational philosophy. This book falls under that category. For example, this book is almost completely devoid of any examples. I don't know about you, but from example, is how I learn. I could go through this book much faster, if there were some decent examples. You can tell me a thousand times what a sigma algebra is, but if you don't give me some decent, worked-out examples which might tell me why tell me why I should learn it (other than because I'll fail the course), I'm going to forget the definition after 5 minutes. Secondly, it would help if there were more pictures. A picture is worth a thousand words. Third, some of the definitions are not worded as well as they should be. last night I spent ten minutes trying to figure out whether the definition for x-section Ex = {y in Y : (x,y) in E} meant that "for all x," or just "for some x?" It turned out it meant "for fixed x." But nowhere was that little tidbit of information written. Ten minutes may not sound like much, but if you have to read 10 pages before you get to pleasure of spending 10 hours with the homework problems, that translates into a lot of time you could spend doing other things if only this book were presented in a manner which would enable you to learn the material more efficiently. I give it two stars primarily because some of the homework problems aren't too bad. If you have a choice, have a look at Kolmogorov and Fomins book on Real analysis. It's not perfect, but the material in it is organized better. (It's not as DENSE) Plus it's a Dover book, and therefore much cheaper.
Rating: 
Summary: TOO MANY TYPOS.
Review: Strictly from a student's perspective, this is a good textbook in real analysis. The way the material is presented is logical, whatever that means, and consistent. The author doesn't assume a student knows much and you can never go wrong with this assumption. However, this book has TOO MANY typos, so if you've never seen this stuff before, it's not easy to follow. Don't trust this book. Hopefully, next edition will fix this problem. This is a good book and it can become a classic, but, first, they have to correct all the typos. For those, who want a really good book in Analysis which has no typos, I recommend Rudin, "Real and Complex Analysis".
Rating: 
Summary: Great book
Review: What I like most about the book is its concise but broad coverage of the fundamentals of real and functional analysis. Although I am not a mathematician, my main interest is solving various engineering problems using numerical methods. A solid background in real and functional analysis would be necessary for deeper understanding of various numerical methods. I wish I had known this book the first time I felt the need to study basic modern analysis. This books has exactly the material I wanted to learn. In addition to the theorems and proofs, the author tells you why some theorems are important and how they can be used (of course also in a concise way). I found this type of "hints" are extremely helpful. The historical notes at the end of each chapter are also interesting to read.
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