Rating: 
Summary: Not for the (even well prepared) student
Review: A thousand years ago I suffered through a real analysis course using Royden's text. This book is clearly NOT written with the student in mind ... in my view it's written to impress Royden's peers. I find little improvement in the current edition and would strongly encourage beginning graduate students in mathematics to examine the many excellent alternatives.Analysis is a fascinating branch of mathematics. This text does not even come close to doing it justice.
Rating:
Summary: Not for the (even well prepared) student
Review: A thousand years ago I suffered through a real analysis course using Royden's text. This book is clearly NOT written with the student in mind ... in my view it's written to impress Royden's peers. I find little improvement in the current edition and would strongly encourage beginning graduate students in mathematics to examine the many excellent alternatives. Analysis is a fascinating branch of mathematics. This text does not even come close to doing it justice.
Rating: 
Summary: I'm surprised too.
Review: All the negative oppinions say that It repeats allmost identical
contents twice. I cannot understand why so many people feel so uneasy about that. Acutally, It does not repeat the "same" thing. In part one, measure theory on the real line is presented and, after you get pretty good understanding and image in the "real world", the abstraction (or equivalently, axiomazation)of measure In abstract space is given. I think It's the best way of explaining something. I know that many people in this field love the "rudin Style" - Books which contains Definitions and Theorems only.
Oh I envy them. I wish I had the ability to understand something from the essense without rumbling the world I can touch for some time. but It's pretty hard thing to do for me, and I'm sure that most others also are. If you agree with the negative oppinions than you can start from part three. you can get all the contents of Part One. and hopefully, You would be able to Understand all the materials. But I don't think that you will get an clearer image than those who have startde from part one.Even those who complain the system of this book could do that cause they already have read the part one and studied the "same thing" Twice. If they have not read them and started from the part three, they would have complained that It was so abstract.
Rating: 
Summary: this book is just plain good.
Review: I began as a graduate student in applied maths less than a year ago; all of the students that I spoke with prior to that said that real analysis with rudin's book was their worse & hardest class..
So when I walked into MTH 5111 Real Variables I thought oh *&^% what am I in for?? but then I picked up the Royden book and I understood the way he was presenting the materail.. the book is very stright to the point + leaves channelgning problems to the HW sets but the autor clearly outlines. I have learned more from this book and course than any other...
Rating:
Summary: this book is just plain good.
Review: I began as a graduate student in applied maths less than a year ago; all of the students that I spoke with prior to that said that real analysis with rudin's book was their worse & hardest class..
So when I walked into MTH 5111 Real Variables I thought oh *&^% what am I in for?? but then I picked up the Royden book and I understood the way he was presenting the materail.. the book is very stright to the point + leaves channelgning problems to the HW sets but the autor clearly outlines. I have learned more from this book and course than any other...
Rating:
Summary: real analysis
Review: I did not find this text as compelling as I thought I would when I bought it. I was amazed at the complete lack of specific examples. While it is true that this causes the reader to develop them himself, one does not usually have that amount of free time, especially while concurrently studying other areas of mathematics, computer science, etc. All in all I guess I will have to wait for the movie.
Rating:
Summary: Not bad for self-study, excellent for reference
Review: I used Royden (2nd edition) as a graduate student over 30 years ago, and have been away from real analysis pretty much ever since (not because of the book(!), but because of being in computers). I've taken a renewed interest in the subject (I'm a pretty random person) and have been surprised at how the material has come back to me, I think because of the readability of the text. It's true, Royden challenges the reader at every turn, but if one has acquired the level of mathematical maturity commensurate with strong interest in analysis, the challenges are appropriate, in my opinion
Rating:
Summary: Not bad for self-study, excellent for reference
Review: I used Royden (2nd edition) as a graduate student over 30 years ago, and have been away from real analysis pretty much ever since (not because of the book(!), but because of being in computers). I've taken a renewed interest in the subject (I'm a pretty random person) and have been surprised at how the material has come back to me, I think because of the readability of the text. It's true, Royden challenges the reader at every turn, but if one has acquired the level of mathematical maturity commensurate with strong interest in analysis, the challenges are appropriate, in my opinion
Rating:
Summary: Excellent Moore Method Intro Text
Review: I used this text as senior undergraduate in an introductory course to real variables. The course was structured as a sort of modified RL Moore method class: there were very few lectures, and we (the students) could only use theorems and propositions presented in the text if we had gone to the board and presented a valid proof for each. As such, most of the students learned the fundamentals very well. This in turn made my first graduate course in real variables much easier. The biggest downside however is that most graduate students don't have the time needed to dedicate to the various problems in this text, which is why Royden is probably not the best choice for a first year graduate text. Instead I would recommend Bartle's Elements of Integration and Lebesgue Measure as a first year grad text on the subject. It was disappointing to use Bartle and discover that so many of the problems in Royden, which I had spent countless hours attempting to prove, had been completely worked out in Elements of Integration. In short, Royden makes you work for many (most?) important results, and in the long run this makes for a much stronger understanding of the material- if you have the time to devote to it.
Rating:
Summary: Excellent Moore Method Intro Text
Review: I used this text as senior undergraduate in an introductory course to real variables. The course was structured as a sort of modified RL Moore method class: there were very few lectures, and we (the students) could only use theorems and propositions presented in the text if we had gone to the board and presented a valid proof for each. As such, most of the students learned the fundamentals very well. This in turn made my first graduate course in real variables much easier. The biggest downside however is that most graduate students don't have the time needed to dedicate to the various problems in this text, which is why Royden is probably not the best choice for a first year graduate text. Instead I would recommend Bartle's Elements of Integration and Lebesgue Measure as a first year grad text on the subject. It was disappointing to use Bartle and discover that so many of the problems in Royden, which I had spent countless hours attempting to prove, had been completely worked out in Elements of Integration. In short, Royden makes you work for many (most?) important results, and in the long run this makes for a much stronger understanding of the material- if you have the time to devote to it.
| 
