Real Mathematical Analysis [Hardcover]

Charles Chapman Pugh (Author)

Book Description

Publication Date: March 1, 2002 | ISBN-10: 0387952977 | ISBN-13: 978-0387952970

Was plane geometry your favourite math course in high school? Did you like proving theorems? Are you sick of memorising integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians like Dieudonne, Littlewood and Osserman. The author has taught the subject many times over the last 35 years at Berkeley and this book is based on the honours version of this course. The book contains an excellent selection of more than 500 exercises.

Editorial Reviews

Review

From the reviews:

C.C. Pugh

Real Mathematical Analysis

"The book contains more than 500 carefully worked exercises the level of which varies from straight-forward to challenging (the text can thus serve as a source book for examples and exercises in real analysis). The book can be highly recommended as an introduction to real analysis."—ZENTRALBLATT MATH

"This introduction to undergraduate real analysis is based on a course taught … by the author over the last thirty-five years at the University of California, Berkeley. … The exposition is informal and relaxed, with a number of pictures. The emphasis is on understanding the theory rather than on formal proofs. The text is accompanied by very many exercises, and the students are strongly encouraged to try them." (EMS Newsletter, June, 2003)

"This one is … a pleasure to read, contains many exercises (about 500) and includes full proofs following an intuitive introduction of new ideas. I should add here that Pugh succeeds in transferring his love and enthusiasm for this material to the reader. … As a mathematics student, I would have loved to have this as a textbook to be taught my first analysis course. As a teacher … I would love to use it as lecture notes." (Adhemar Bultheel, Belgian Mathematical Society - Simon Stevin Bulletin, Vol. 11 (1), 2004)

"The author of this undergraduate text believes that real analysis is the jewel in the crown of pure mathematics. … This text is based upon many years of teaching the analysis course at Berkeley. The exposition is chatty and easygoing, while managing to cover all of the basic ideas carefully and thoroughly. … The text is complemented by an excellent index and frequent suggestions for further reading. I can recommend this book to serious undergraduates who want to get into real analysis … ." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (551), 2004)

"This book is a new introduction to undergraduate real analysis. … The exposition is informal and relaxed, with an excellent selection of more than 500 exercises. The occasional comments from mathematicians … make the text really enjoyable. … To sum up, this undergraduate … textbook contains a wealth of information. It is written in a concise, but always clear and well-readable way. … It should have a great appeal to the students of (under) graduate courses as well as to budding pure mathematicians." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 69, 2003)

"Pugh’s book … is not a typical book. … it very successfully (and atypically) manages to convey the look and feel of an engaging classroom lecture while maintaining the highest level of rigor and care. … This makes this well-crafted book very appealing as a resource for an honors section, but it also should be in any undergraduate library as a source of ideas and supplementary problems for faculty or as a challenge for strong students. An excellent book in an excellent series. Highly recommended." (J. Feroe, CHOICE, September, 2002)

"This book is suited for a two-semester course in real analysis for upper-level undergraduate students who major in mathematics. … The book is very well written. The style is lively and engaging. Intuition is stimulated and metaphors are used throughout the book, without compromising rigor. … The exercises are numerous and they vary from straightforward to very challenging … . This is a book for the highly motivated student. He/she will get from this book a good grasp of analysis: concepts and techniques." (Sherif T. El-Helaly, Mathematical Reviews, 2003 e)

"The book under review is an introduction to the basics of real analysis. … A special feature of the exposition is its emphasis on the explanation of mathematical concepts by figures … . The book can be used for self-study. … The book can be highly recommended as an introduction to real analysis." (Joachim Naumann, Zentralblatt MATH, Vol. 1003 (3), 2003)

"In this new introduction to undergraduate real analysis, the author takes a different approach … by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples, and occasional comments from mathematicians … . This book is based on the honors version of a course that the author has taught many times, over the last thirty-five years, at the University of California, Berkeley." (L’ Enseignement Mathematique, Issue 1-2, 2002)

From the Back Cover

Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician.
In this new introduction to undergraduate real analysis the author takes a different approach from past presentations of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians such as Dieudonne, Littlewood and Osserman. This book is based on the honors version of a course which the author has taught many times over the last 35 years at Berkeley. The book contains an excellent selection of more than 500 exercises.

Product Details

• Hardcover: 456 pages
• Publisher: Springer (March 1, 2002)
• Language: English
• ISBN-10: 0387952977
• ISBN-13: 978-0387952970
• Product Dimensions: 9.3 x 6.1 x 1 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (14 customer reviews)
• Amazon Best Sellers Rank: #203,106 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

70 of 73 people found the following review helpful:

5.0 out of 5 stars Very good exposition, great problems, April 3, 2004

By 

David Hagar (Berkeley, CA) - See all my reviews

This review is from: Real Mathematical Analysis (Hardcover)

Real analysis is a genre with an established classic (Rudin) and a plethora of available books and resources. Unfortunately, most analysis books cost a great deal of money so the average reader will only purchase one or two texts. In evaluating which book(s) to purchase two questions should be asked:

1.) Why purchase this book rather than the classic of the genre?

2.) Is this book appropriate for me?

So why buy this book rather than Rudin? It has great exposition (as does Rudin), very well chosen problems (as does Rudin), but Pugh manages to improve on the standard by supplementing his written explanations with diagrams and pictures that Rudin mostly lacks. Additonally, the price stands at something less than half the cost of Rudin's book.

Who is this book appropriate for? This text delves into the topological underpinnings of analysis. It is not an "analysis-lite" textbook a la Ken Ross's Elementary Analysis. It is a rigorous treatment of the subject, and it has a comprehensive feel to it, covering topics like Lebesgue measure and integration, and multivariable analysis in addition to the normal topics one would expect. In short, it is appropriate for somebody who is seeking the challenges and rewards of a full treatment of what for many is a difficult subject.

It is a very good book that does not shy away from difficult material that no amount of explanation or good writing will make easy to learn, but of all the analysis books I've seen, this comes the closest.

49 of 50 people found the following review helpful:

5.0 out of 5 stars Improves on the classic, May 27, 2006

By 

C. Dolan (New York, NY) - See all my reviews
(REAL NAME)   

This review is from: Real Mathematical Analysis (Hardcover)

As a previous reviewer has noted, Walter Rudin's Principles of Mathematical Analysis is the standard textbook for a rigorous analysis course. Rudin's book is very good because of the level of rigour and abstraction, the bredth of material covered, the way it forces the reader to fill in the blanks, and because of the challenging exercises throughout. In my opinion, Pugh has managed to improve on the classic in every aspect.

First of all, he does not develop all the concepts in same order as Rudin - first he develops the real number system, a few basic things about Cauchy sequences, and then moves onto continuity. Then he goes into a lengthy chapter on topology, which, in my humble opinion, is where the book first outshines Rudin. He defines compactness in terms of the convergence of subsequences, which is much more natural than the covering definition. He later proves that the two conditions are equivalent. In the third chapter, he develops differentiation and integration, much in the way Rudin does. In the fourth chapter, develops series and sequences (of functions). In the fifth chapter, he develops multivariable calculus, and the in the sixth chapter, he develops measure theory and the Lebesgue integral. Since there are fewer chapters than there are in Rudin's book, I think he develops the subject matter in a more natural, cohesive manner.

Rudin's book is excellent through the series and sequences of function. It is generally agreed that the book tails off after the seventh chapter, that is, he does not do as good a job with multivariable calculus and Lebesgue Theory. Pugh manages to do a good job throughout, so in addition to having a better chapter in topology, he is better than Rudin in those areas. I also believe that his treatment of series and sequences of functions is more interesting: Rudin treats them, for the most part, as distinct mathematical objects, and only briefly makes reference to the space of functions, whereas Pugh centers the chapter around the idea of function spaces (the heart of real analysis, really). Furthermore, Pugh uses illustrations (not too many, but enough) to illustrate certain concepts, and in fact, to simplify certain proofs. He also emphasizes the utility of geometric thinking in developing proofs, something which Rudin does not do. Furthermore, Rudin is notoriously terse; I think Pugh does a better job motivating and explaining the material without being "chatty" (the cardinal sin in mathematical exposition), while not insulting the reader's intelligence, that is, you are expected to fill in certain gaps on your own.

I would also like to emphasize the quality of the exercises in this book. There are many, many exercises - more than PMA, in fact. None of them are trivial. Many of them are quite challenging, on par with those in Rudin's book. Unlike Rudin, though, Pugh includes a fair share of easier, but still interesting exercises, which I think are essential for really getting a grasp on the material. He also has some problems, I think, which are a good bit harder than any of Rudin's, which is saying a lot, so there is something for everyone here.

Overall, I think this is the best book out there for an intro to analysis course. The price is also quite reasonable, considering how much math books tend to cost.

20 of 21 people found the following review helpful:

5.0 out of 5 stars excellent, June 6, 2003

By 

Dima (United States) - See all my reviews

This review is from: Real Mathematical Analysis (Hardcover)

This is one of the best books on introductory real analysis that I have looked at. Before I found this book, I have been reading another work on real analysis which was also very good, but was far less comprehensive. Not only does this book present a precise exposition of concepts and theorems, it also gives illustrations to better explain the ideas and plenty of excercises at the end of each chapter. For example, the author does not only say what a "covering" means, but he gives an illustration of it. The style of exposition is fine and relaxed, but the rigor of presentation of theorems and proofs is not in the least compromised. I would think that this book will be of enormous help to anyone trying to make a transition from concrete to more abstract mathematical reasoning.