mathematics

I love this book. Its difficulty is legendary, yet that's also its charm.

Certainly, years after taking a course with this book, you can look back at it and it doesn't seem so hard. And many books in more advanced subjects will, naturally, seem harder. But relative to the situation you are in when you first learn the material (in this case right after the usual lower-division sequence of calculus, linear algebra and differential equations), this may be the hardest math book you ever encounter.

I used Rudin's book in my first course in analysis during my sophomore year at UC Berkeley and, amazingly, lived to tell about it. This book was a shock to the system. I spent ungodly amounts of time trying to figure out Rudin's insanely terse proofs, and even more time on the *hard* exercises.

Rudin covers all the material that provides a foundation for subsequent analysis courses, in Chapters 1-8. His conciseness turns off many people, but I felt it helped me learn the material better because it *forced* me to read every single word and actually *study* what he wrote. You can not just skim through this book.

The first 8 chapters are so good that it excuses the lesser quality of the last 3 chapters. The multivariable calculus material in Chapters 9-10 is not that strong (see Edwards' "Advanced Calculus of Several Variables" for a better treatment). And Lebesgue theory (Ch.11) is better left for a more advanced book (e.g. Royden's "Real Analysis").

This is a serious, no-nonsense math book. If you do not want to be challenged, then you should use a different book. Back when I took analysis, other books used were Marsden's "Elementary Classical Analysis", Bartle's "Intro. to Real Analysis", and Protter & Morrey's "A First Course in Real Analysis", all of which cover a lot of the same material as Rudin but "friendlier".

Most reviewers recommend using a different book for an intro to analysis, while only a few recommend this as a first book. My recommendation would be:
a) If you want to be a mathematician then definitely use Rudin. This is where you will learn how to write formal proofs, and the style will prepare you for more advanced study. This book introduces you to the language of professional mathematicians.
b) If you are not a math major, then use Rudin only if you are extremely strong in math (e.g. you were able to do proofs easily in your linear algebra class, theoretical exercises didn't scare you). Otherwise, I would recommend the above-mentioned books by (in order of the most difficult) Marsden, Protter & Morrey, or Bartle.

Rudin's elementary analysis text is simply the best. This is mostly attributable to its very precise and compact language. (I wish all proofs were written like this.) The book is easy to read and largely understandable. It makes good first time reading as well as a handy reference.

(By first time reading, I don't mean that the text is recommended for people without formal exposure to proofs--Rudin's text simply assumes that one is already familiar with the rules and concepts of basic logic. On the other hand, if one has already had some exposure to proofs in a non-analysis context (i.e. number thoery), then Rudin's book can serve as a very good intro text to analysis.)

Many other analysis texts are simply too verbose, taking paragraphs and paragraphs to describe concepts that can be easily/better said in just a few words. As a result they can't be read in a feasible amount of time. I also dislike texts that are not written in the definition-axiom-proposition-proof manner--I find that to be a lack of mathematical discipline and consistency. And I doubt that one can gain much mathematical maturity by reading texts of this sort.

If one wants to seek mathematical motivation behind real analysis or a starter's book that introduces basic logic, something that the Rudin text clearly lacks, I *highly* recommend "The Lanaguage of Mathematics" by Keith Devlin. This is a popular math book that is suitable for all levels, and it successfully communicates the basic ideas and motivation behind many branches of math. It also covers most of the important historical development of classical mathematics with many engaging stories. Another book that I can really recommend as a more understandable substitute for Rudin's text is "Elementary Mathematical Analysis" by Colin W. Clark. This book is the most understandable intro analysis text I've seen and contains sufficient coverage of logic and mathematical motivation to be a standalone text. However, it's no longer in print but should be available at most school's libraries.

This book is a standard undergrad. introduction to Analysis. It provides a nice foundation, making you work at reading proofs and solving problems while getting familiar with the basic concepts -- limsups and infs, basics of continuity, compactness, etc. You would perhaps be better served if this using this book is not your first experience with really doing mathematics, e.g. formal proofs, etc. -- though not Spivak's Calculus on Manifolds or one of J.P. Serre's Arithmetic books, this book is more concise than many. Important theorems such as the Stone Weierstrass are proven in a very clean brief way (this may not lead to the most useful of proof styles -- you may find yourself expending precious time on cleaning up proofs -- "does leaving this step in make me look stupid?" -- and perhaps cutting so much that proofs may look "infelicitous."). I also do not remember this book being strong on Lebesgue theory and don't remember discussion of Littlewood's principles, Radon Nikodym, etc. These, the real substance of Real Analysis, are best seen in Royden or Rudin's Real and Complex book.Moreover, some professors prefer the sigma algebra approach to measures -- the wonderful S. Kakutani, for example, who briefly guest taught the class in which I used this book insisted on reteaching measures using sigma algebras.