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12 people found this helpful

ByPalle E T Jorgensenon 21 September 2004

I am a fan of Rudin's books. This one "Principles of Matheamtical Analysis" has served as a standard textbook in the first serious undergraduate course in analysis at lots of universities in the US, and around the world.

The book is divided in the three main parts, foundations, convergence, and integration. But in addition, it contains a good amount of Fourier series, approximation theory, and a little harmonic analysis.

I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.

What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.

The exercises and just at the right level. They can be assigned in class, or students can work on them alone. I think that is good, and the exercises serve well as little work-projects. This approach to the subject is probably is more pedagogical as well.

I think students will learn things that stay with them for life.

Review by Palle Jorgensen.

The book is divided in the three main parts, foundations, convergence, and integration. But in addition, it contains a good amount of Fourier series, approximation theory, and a little harmonic analysis.

I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.

What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.

The exercises and just at the right level. They can be assigned in class, or students can work on them alone. I think that is good, and the exercises serve well as little work-projects. This approach to the subject is probably is more pedagogical as well.

I think students will learn things that stay with them for life.

Review by Palle Jorgensen.

15 people found this helpful

ByA customeron 21 July 1999

I was terrified by this text my freshman year in college. Unfortunately, this book was the only required book for the class. The main difficulty is that the book resembles a magnificent outline of the material more than a text. The shortest, most elegant proof of anything is invariably choosen and there is little motivation given for the material. Thus, I found this book to be difficult to use to learn how to do mathematics. On the other side, if you know the basic ideas of analysis, then this book is a remarkable, clear, and elegant place to review and extend your knowledge. I therefore would HIGHLY recommend it as a companian text for an analysis course or as a reference. My rating therefore is an average: as an introduction to analysis by itself, it rates one star; as a supplement to another text, as a review text, or as a reference, it clearly rates five stars.

ByA customeron 21 July 1999

I was terrified by this text my freshman year in college. Unfortunately, this book was the only required book for the class. The main difficulty is that the book resembles a magnificent outline of the material more than a text. The shortest, most elegant proof of anything is invariably choosen and there is little motivation given for the material. Thus, I found this book to be difficult to use to learn how to do mathematics. On the other side, if you know the basic ideas of analysis, then this book is a remarkable, clear, and elegant place to review and extend your knowledge. I therefore would HIGHLY recommend it as a companian text for an analysis course or as a reference. My rating therefore is an average: as an introduction to analysis by itself, it rates one star; as a supplement to another text, as a review text, or as a reference, it clearly rates five stars.

ByPalle E T Jorgensenon 21 September 2004

I am a fan of Rudin's books. This one "Principles of Matheamtical Analysis" has served as a standard textbook in the first serious undergraduate course in analysis at lots of universities in the US, and around the world.

The book is divided in the three main parts, foundations, convergence, and integration. But in addition, it contains a good amount of Fourier series, approximation theory, and a little harmonic analysis.

I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.

What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.

The exercises and just at the right level. They can be assigned in class, or students can work on them alone. I think that is good, and the exercises serve well as little work-projects. This approach to the subject is probably is more pedagogical as well.

I think students will learn things that stay with them for life.

Review by Palle Jorgensen.

The book is divided in the three main parts, foundations, convergence, and integration. But in addition, it contains a good amount of Fourier series, approximation theory, and a little harmonic analysis.

I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.

What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.

The exercises and just at the right level. They can be assigned in class, or students can work on them alone. I think that is good, and the exercises serve well as little work-projects. This approach to the subject is probably is more pedagogical as well.

I think students will learn things that stay with them for life.

Review by Palle Jorgensen.

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ByA customeron 4 September 1999

For years, Dr. Rudin's book has been a standard introductory analysis text because of its wonderful, elegant exposition. It is not an easy book. It was never intended to be an easy book. But those who complain that the book lacks pictures are missing the point. The point of the book is to LEARN analysis. Rudin's book is excellent for this: you have to understand the theorems, definitions, and proofs, otherwise it's nonsense. If one takes the time to understand how all the statements follow from each other, then one will have truly learned analysis, and that is really the point.

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ByA customeron 5 September 1999

Please don't "learn it yourself" from rudin. I recommend books by Bryant, Stirling, and Eccles for the learning of analysis. Those are user friendly books with nice explainations. Only use Rudin once you have learned some analysis and proof. Moreover, the price is a joke. Rudin will be at your library so I recommend borrowing before buying. The other reviewers who gave it such bad reviews are probably mad because they didn't use these other user friendly books first and then switch to rudin. thank you.

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ByA customeron 1 January 1999

this is the book which introduced me to "real" mathematics.i started reading it when i was in the second year of undergraduation and from that moment it became one my favourites (along with herstein's book on algebra and simmons' book on general topology). this book makes you think.(i still remember how that excercise "is there any nonempty perfect set with no rationals?" gave me sleepless nights!)

Dr.Rudin! thank you for giving me such a nice introduction of mathematical analysis.

Dr.Rudin! thank you for giving me such a nice introduction of mathematical analysis.

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ByA customeron 23 December 1998

I used Rudin in my freshman analysis class as an undergraduate. I found it fairly difficult to read. It seems as though Rudin fine tuned his proofs mostly for brevity rather than clarity. For a young student of analysis I think it is better to see the ideas and structures rather than to be forced to decipher overly short proofs. On the other hand - I did learn the material pretty well. Still, I would not reccomend this as a first book and certainly not as a do-it-yourself.

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BySilvio C.on 5 July 2011

This book is popularly called the Little Rudin, compared to the Big Rudin which deals with more advanced topics in mathematical analysis. Thought as a textbook for an undergraduate course, it likely spans two years in a student's career, introducing to all the basic topics of univariate and multivariate calculus, together with complex analysis and formal power series, and even more specialized topics such as Euler's gamma function. The text is always clear and the proof are well constructed; some topics are put in the exercises sections, which forces the reader to actually go through them, a choice which I cannot object to. This book should have a place on the bookshelf of everyone who holds mathematics as a strong interest.

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ByWarwick/IC Maths graduateon 13 December 2010

If not an introductory text, it is certainly a great review text.

1st year mathematics is heavily focused on grounding the student in mathematical analysis. Sequences, series, limits, sums, continuity and differentiability are the bread & butter concepts every maths student needs to know inside out. They call this task, 'the leap from school maths into university maths'.

Sets, functions, proofs, linear algebra, groups and permutations are the foundations of the 1st year but they are quite readily taken on board by most new students because they are in a sense more familiar and intuitive.

If anything is going to seem unfamiliar to a fresher it will be analysis but by the end of your degree, you will come to regard this stuff to be as fundamental to your maths degree as counting.

I found myself using this book quite a lot both in my 1st and 2nd year. By the 2nd year, it tends to replace all your first year analysis notes because it is all concisely laid out in this book for your review.

ps: I really like the section about differential forms personally near the end of the book.

1st year mathematics is heavily focused on grounding the student in mathematical analysis. Sequences, series, limits, sums, continuity and differentiability are the bread & butter concepts every maths student needs to know inside out. They call this task, 'the leap from school maths into university maths'.

Sets, functions, proofs, linear algebra, groups and permutations are the foundations of the 1st year but they are quite readily taken on board by most new students because they are in a sense more familiar and intuitive.

If anything is going to seem unfamiliar to a fresher it will be analysis but by the end of your degree, you will come to regard this stuff to be as fundamental to your maths degree as counting.

I found myself using this book quite a lot both in my 1st and 2nd year. By the 2nd year, it tends to replace all your first year analysis notes because it is all concisely laid out in this book for your review.

ps: I really like the section about differential forms personally near the end of the book.

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ByA customeron 9 March 1999

This book does not have a single picture or illustration in it. That is rare for a mathematical textbook (particularly analysis). You are forced, and rightly so, to form your own mental images of the mathematical objects defined and constructed within the text. This book is logical. Rudin lays it out in the definition, theorem, proof format, but does it in an amazing way. He leads you through a series of minor theorems and lemmas, you have no idea where he is going (unless you've already studied analysis) but then it all leads to a major result (for example the Heine-Borel Theorem in the Topology chapter) which is then used in the proof of many other theorems. Typical of how the book is written, he never tells you how important things like the Heine-Borel Theorem are, but the astute student soon figures it out. He does occassionally give a sentence or two of explanation or elucidation, but he mostly leaves that to the professor teaching the course. The exercises are excellent; tough and illuminating. Do as many as you can and you will learn a lot. If you can handle it, Rudin is the best way to learn Analysis (i.e., no BS). Good backround material before tackling Rudin would be Spivak's "Calculus" or Courant/John's "Introduction to Calculus and Analysis."

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ByA customeron 15 October 2001

I was recommended this book for my maths degree and I can honestly say that it's really useful. A good broad overview of analysis, with lots of exercises to assist your understanding of unfamiliar concepts. Starting with basic notions of metric and building upon them, through algebraic systems, differentiation, integration, partial differentiation and finishing with a small chapter on the Lebesgue integral.

My only real gripe with this book was that it did not have enough operator theory in it. (I would recommend Simmons 'Intro to topology and modern analysis' (same series) or Kreyzig's 'Intro to functional analysis' (Wiley Classics) for operator theory.)

My only real gripe with this book was that it did not have enough operator theory in it. (I would recommend Simmons 'Intro to topology and modern analysis' (same series) or Kreyzig's 'Intro to functional analysis' (Wiley Classics) for operator theory.)

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