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.
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.
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.
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.
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.