This course in real analysis is directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something of value to specialists and nonspecialists alike. The text covers three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal, down-to-earth style, the author gives motivation and overview of new ideas, while still supplying full details and complete proofs. He provides a great many exercises and suggestions for further study.
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Editorial Reviews
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'... extremely well written: very entertaining and motivating.' Adhemar Bultheel, Bulletin of the London Mathematical Society
'The author writes lucidly in a friendly, readable style and he is strong at motivating, anticipating and reviewing the various themes that permeate the text ... The overwhelming impression is that Real analysis was a labour of love for the author, written with a genuine reverence for both its beautiful subject matter and its creators, refiners and teachers down the ages. As such - and high praise indeed - it will sit very happily alongside classics such as Apostol's Mathematical analysis, Royden's Real analysis, Rudin's Real and complex analysis and Hewitt and Stromberg's Real and abstract analysis.' The Mathematical Gazette
'The author writes lucidly in a friendly, readable style and he is strong at motivating, anticipating and reviewing the various themes that permeate the text ... The overwhelming impression is that Real analysis was a labour of love for the author, written with a genuine reverence for both its beautiful subject matter and its creators, refiners and teachers down the ages. As such - and high praise indeed - it will sit very happily alongside classics such as Apostol's Mathematical analysis, Royden's Real analysis, Rudin's Real and complex analysis and Hewitt and Stromberg's Real and abstract analysis.' The Mathematical Gazette
Book Description
This is a course in real analysis directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something to specialists and non-specialists alike, including historical commentary, carefully chosen references, and plenty of exercises.
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Most Helpful Customer Reviews
12 of 13 people found the following review helpful
Format:Paperback|Amazon Verified Purchase
In the author's preface, he states that the prerequisites are "one semester of advanced calculus or real analysis at the undergraduate level". So, this book cannot be judged as an 'intro to real analysis'.
I just want to comment on how I have experienced this book. Let me mention that I am using this for self-study after completing a course using Rudin's Principles of Mathematical Analysis (we covered every chapter except Ch. 10 on integration in R^n). I picked this up to review analysis with the goal of covering function spaces and measure theory with more emphasis that Rudin. This book does just that! But, I also wanted a book that stays in R for the Lebesgue measure. Having read the first 3 chapters of Folland, I didn't really think I 'understood' the material even though I could do the exercises (but not without a lot of sweat and coffee). (At one point I felt I became a function: [input] facts, assumptions then [output] proofs, ie hw exercises.) Folland does everything for abstract measures and treats the Lebesgue measure as a corollary.
Having said that, this books hits the spot.
A previous reviewer said this book was informal, unprofessional, and chatty. I do agree with him on that the book is very informal in the exposition and is chatty. I feel that this might be very distracting for those who do not wish to be specialists in analysis, or to those who are seeing analysis for the first time. However, for someone who has finished, say Baby Rudin, this book IS AMAZING. His chatty 'foreshadowing' is the best part, since by now you are trying to see the 'big picture'. In this respect, the chattiness tells of the shortcomings of the previous theory and points one to the right questions to ask.
I think this book shines for the purpose of an intermediate course between Baby Rudin and graduate real analysis ala Folland. As such, the exercises are at the perfect level and include standard, important, and interesting results and extensions. I don't think this book is rigorous enough for a real course at the graduate level, however.
A final note, the editorial (why?)'s placed throughout do get annoying but I feel they make sure you do not take results for granted, an all too common habit when reading advanced math.
I just want to comment on how I have experienced this book. Let me mention that I am using this for self-study after completing a course using Rudin's Principles of Mathematical Analysis (we covered every chapter except Ch. 10 on integration in R^n). I picked this up to review analysis with the goal of covering function spaces and measure theory with more emphasis that Rudin. This book does just that! But, I also wanted a book that stays in R for the Lebesgue measure. Having read the first 3 chapters of Folland, I didn't really think I 'understood' the material even though I could do the exercises (but not without a lot of sweat and coffee). (At one point I felt I became a function: [input] facts, assumptions then [output] proofs, ie hw exercises.) Folland does everything for abstract measures and treats the Lebesgue measure as a corollary.
Having said that, this books hits the spot.
A previous reviewer said this book was informal, unprofessional, and chatty. I do agree with him on that the book is very informal in the exposition and is chatty. I feel that this might be very distracting for those who do not wish to be specialists in analysis, or to those who are seeing analysis for the first time. However, for someone who has finished, say Baby Rudin, this book IS AMAZING. His chatty 'foreshadowing' is the best part, since by now you are trying to see the 'big picture'. In this respect, the chattiness tells of the shortcomings of the previous theory and points one to the right questions to ask.
I think this book shines for the purpose of an intermediate course between Baby Rudin and graduate real analysis ala Folland. As such, the exercises are at the perfect level and include standard, important, and interesting results and extensions. I don't think this book is rigorous enough for a real course at the graduate level, however.
A final note, the editorial (why?)'s placed throughout do get annoying but I feel they make sure you do not take results for granted, an all too common habit when reading advanced math.
11 of 12 people found the following review helpful
Format:Paperback
I used this book for a semester course in Analysis II. We didn't read the book in a linear fashion from start to finish, but we managed to thoroughly cover the material first on Banach spaces, then functions of bounded variation, then Stieltjes integration, then Lebesgue measure.
The book's biggest asset is that the majority of its many problems are worth attempting. He scatters them throughout each chapter instead of lumping them all at the end which presumably is more pedagogically sound. I was able to do most of the problems I attempted but not some. I really cannot overstate how good the exercises here are.
Also, Carothers will not hold you by the hand - he inserts a parenthetical "why?" everytime he skips over a detail. I agree with this approach but I think the "why's" ought to be omitted since that one should actively read math is implicit, so such parenthetical remarks are superfluous (cf. Rudin).
When I was taking the course, I said the book was too chatty, but I recant this now. Carothers includes extensive historical commentary when appropriate, which is a refreshing departure from monotony, and enlightening in its own right.
The one drawback to this book is that everything is done on the line R^1. Nonetheless it's done well and thoroughly.
Carothers' book is definitely different from most introductory analysis texts, so I wouldn't expect all students or professors to like it as it's admittedly somewhat idiosyncratic, but ultimately it's first-rate. Moreover, it's only a third of the price of certain canonical introductory analysis books that it may even better.
The book's biggest asset is that the majority of its many problems are worth attempting. He scatters them throughout each chapter instead of lumping them all at the end which presumably is more pedagogically sound. I was able to do most of the problems I attempted but not some. I really cannot overstate how good the exercises here are.
Also, Carothers will not hold you by the hand - he inserts a parenthetical "why?" everytime he skips over a detail. I agree with this approach but I think the "why's" ought to be omitted since that one should actively read math is implicit, so such parenthetical remarks are superfluous (cf. Rudin).
When I was taking the course, I said the book was too chatty, but I recant this now. Carothers includes extensive historical commentary when appropriate, which is a refreshing departure from monotony, and enlightening in its own right.
The one drawback to this book is that everything is done on the line R^1. Nonetheless it's done well and thoroughly.
Carothers' book is definitely different from most introductory analysis texts, so I wouldn't expect all students or professors to like it as it's admittedly somewhat idiosyncratic, but ultimately it's first-rate. Moreover, it's only a third of the price of certain canonical introductory analysis books that it may even better.
2 of 3 people found the following review helpful
Format:Hardcover
I wish I had reviewed this REMARKABLE textbook 5 years ago when I had a lot of knowledge at my fingertips. Now I am not able to go "deep" anymore, which means you may not find this review (that much) helpful. Nevertheless, I am just happy to improve its overall rating (should have done long ago!)
What I still remember though is:
1) what appealed to me most: It was the COHERENCE of the material presented as well as the WAY it was presented. This, coupled with a very good choice of format (size, quality of paper, font, etc.), made it one of the MOST ENJOYABLE learning experiences for me!
2) the EXERCISES being VERY GOOD and interesting (no answers, but some of them had helpful hints).
3) the "WHY?s" thrown in the proofs. I liked this feature a lot. After all, when doing mathematics (proofs), you are not supposed to take anything for granted; on the contrary, you need to justify each and every statement/implication you make. This will make you think about things you considered "basic" or "obvious."
4) the author's PASSION for the subject being felt throughout the book, and being very CONTAGIOUS.
Lastly, let me share a couple of things with you:
1) The REAL ANALYSIS course I took back then was a 2-semester course. I took each semester at different place (within the University), with different professor and using different texbook. While I found the classes in the first semester lacking structure, the second semester made up for it BIG TIME!
2) I have always loved just about anything and everything about mathematics. That being said, I still had my FAVORITEs during the studies, and I can tell you real analysis was not among them. As many of you may guess, this has changed by the time I graduated...
I attribute both things to nothing else but THE TEXTBOOK used (and, of course, the professor and the right choice he made).
HIGHLY RECOMMENDED!
What I still remember though is:
1) what appealed to me most: It was the COHERENCE of the material presented as well as the WAY it was presented. This, coupled with a very good choice of format (size, quality of paper, font, etc.), made it one of the MOST ENJOYABLE learning experiences for me!
2) the EXERCISES being VERY GOOD and interesting (no answers, but some of them had helpful hints).
3) the "WHY?s" thrown in the proofs. I liked this feature a lot. After all, when doing mathematics (proofs), you are not supposed to take anything for granted; on the contrary, you need to justify each and every statement/implication you make. This will make you think about things you considered "basic" or "obvious."
4) the author's PASSION for the subject being felt throughout the book, and being very CONTAGIOUS.
Lastly, let me share a couple of things with you:
1) The REAL ANALYSIS course I took back then was a 2-semester course. I took each semester at different place (within the University), with different professor and using different texbook. While I found the classes in the first semester lacking structure, the second semester made up for it BIG TIME!
2) I have always loved just about anything and everything about mathematics. That being said, I still had my FAVORITEs during the studies, and I can tell you real analysis was not among them. As many of you may guess, this has changed by the time I graduated...
I attribute both things to nothing else but THE TEXTBOOK used (and, of course, the professor and the right choice he made).
HIGHLY RECOMMENDED!
Most Recent Customer Reviews
I hope it's not just me...
I've gone through 1 term of this book and it's been a horrible experience. Before I continue, I should say that I have gone through an entire semester of analysis at the... Read more
Published 6 months ago by ForAllThereExists
Very Good Book
I highly recommend this text to learn from. This book is like taking a course from your favorite teacher, who loves the subject and knows it very well. Read more
Published 22 months ago by Indikos
Sales Rank is Depressingly Low
It is sad to see the sales rank of this book so low, currently
700,000+. The author does a good job explaining real analysis.
This book goes well with R.P. Burn's book.
700,000+. The author does a good job explaining real analysis.
This book goes well with R.P. Burn's book.
Published on August 14, 2003
Very good introduction to Real Analysis!!!
Very good introduction to Real Analysis!!! I highly recommend it to those starting to study the subject.
Published on November 6, 2001 by raymund aglipay
Inside This Book
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First Sentence:
Our goal in this chapter is to provide a quick review of a handful of important ideas from advanced calculus (and to encourage a bit of practice on these fundamentals). Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
trig polynomial, nested set theorem, nested interval theorem, proper closed subalgebra, generic metric space, integrable simple functions, first category set, second category set, polygonal function, least upper bound axiom, finite outer measure, refinement integral, outer measure zero, increases pointwise, nonmeasurable set, nonnegative measurable functions, many closed sets, totally bounded set, pairwise disjoint intervals, disjoint measurable sets, vibrating string problem, pointwise bounded, total boundedness, lim infra, finite derivative Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Corollary, Dominated Convergence Theorem, Monotone Convergence Theorem, The Ohio State University, Other Definitions, Axiom of Choice New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Our goal in this chapter is to provide a quick review of a handful of important ideas from advanced calculus (and to encourage a bit of practice on these fundamentals). Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
trig polynomial, nested set theorem, nested interval theorem, proper closed subalgebra, generic metric space, integrable simple functions, first category set, second category set, polygonal function, least upper bound axiom, finite outer measure, refinement integral, outer measure zero, increases pointwise, nonmeasurable set, nonnegative measurable functions, many closed sets, totally bounded set, pairwise disjoint intervals, disjoint measurable sets, vibrating string problem, pointwise bounded, total boundedness, lim infra, finite derivative Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Corollary, Dominated Convergence Theorem, Monotone Convergence Theorem, The Ohio State University, Other Definitions, Axiom of Choice New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 81
books:
See all 81 books this book cites
2
books
cite this book:
- Mathematical Analysis by Wiley on 4 pages
- The World of Mathematics by James R. Newman in Back Matter (1), and Back Matter (2)
- Studies in Harmonic Analysis (Mathematical Association of America. Studies in Mathematics, Vol 13) by J. Marshall Ash in Back Matter (1), and Back Matter (2)
- Studies in Modern Analysis by Robert Creighton Buck in Back Matter (1), and Back Matter (2)
- Set Theory. Third Edition by Felix Hausdorff on page 50, and Back Matter
See all 81 books this book cites
- The Theory of Measures and Integration (Wiley Series in Probability and Statistics) by Eric M. Vestrup in Back Matter
- Basic Real Analysis by Houshang H. Sohrab in Back Matter
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