- Paperback: 416 pages
- Publisher: Cambridge University Press; 1 edition (August 15, 2000)
- Language: English
- ISBN-10: 0521497566
- ISBN-13: 978-0521497565
- Product Dimensions: 7 x 0.9 x 10 inches
- Shipping Weight: 2 pounds (View shipping rates and policies)
- Average Customer Review: 4.6 out of 5 stars See all reviews (14 customer reviews)
- Amazon Best Sellers Rank: #708,070 in Books (See Top 100 in Books)
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Real Analysis 1st Edition
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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
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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Top Customer Reviews
This book is like taking a course one who loves the subject and knows it very well.
And what is more, she wants to communicate her knowledge to you.
She has anticipated your difficulties and is prepared to guide you.
The material is well motivated; the historical asides help you understand and appreciate the material even more.
There are a lot good exercises.
One caveat is that analysis is on R not R^n.
The material is pitched at the senior level.
I wish I had this book when I was studying this subject.
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.
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.
This book is great if you've had a reasonable real analysis course and want to review what you've learned while simultaneously going into a little more depth or if you want to teach yourself basic Lebesgue theory. For the former, its coverage of function spaces and topics like Stone-Weierstrass is excellent (imagine Ch. 7 of Baby Rudin expanded to several chapters). For the latter, it offers the most intuitive and systematic (step-by-step) development of the Lebesgue integral I've seen. This book is worth getting just for that part. Also, if you're really smart, this book is gentle enough to conceivably be used as a first text in real analysis, provided you've already seen epsilon-delta, difference quotients, Riemann sums, etc. in a rigorous calculus book (Apostol or Spivak), or don't care if you learn things "out of order".