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[Nadia Summerhill]

Doesanyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.

While it's a good book, I'm not sure it's suited for self study by itself. I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway.

When I was learning introductory real analysis, the text that I found the most helpful was Stephen Abbott's Understanding Analysis. It's written both very cleanly and concisely, giving it the advantage of being extremely readable, all without missing the formalities of analysis that are the focus at this level. While it's not as thorough as Rudin's Principles of Analysis or Bartle's Elements of Real Analysis, it is a great text for a first or second pass at really understanding single, real variable analysis.

If you're looking for a book for self study, you'll probably fly through this one. At that point, attempting a more complete treatment in the Rudin book would definitely be approachable (and in any case, Rudin's is a great reference to have around).

For self-study, I'm a big fan of Strichartz's book "The way of analysis". It's much less austere than most books, though some people think that it is a bit too discursive. I tend to recommend it to young people at our university who find Rudin's "Principle of mathematical analysis" (the gold standard for undergraduate analysis courses) too concise, and they all seem to like it a lot.

EDIT : Looking at your question again, you might need something more elementary. A good choice might be Spivak's book "Calculus", which despite its title really lies on the border between calculus and analysis.

Bryant [1] would be my recommendation if you're fresh out of the calculus/ODE sequence and studying on your own. If your background is a little stronger, then Bressoud [2] might be better. Finally, you should take a look at Abbott [3] regardless, as I think it's the best written introductory real analysis book that has appeared in at least the past couple of decades.

You might want to take a look at A Problem Text in Advanced Calculus by John Erdman. It's free, well-written and contains solutions to many of the exercises. These attributes, in my opinion, make it particularly well-suited for self-study. One of the things that I particularly like about the text is the author's use of o-O concepts to define differentiability. It simplifies some proofs dramatically (e.g., the Chain Rule) and is consistent across one-dimensional and n-dimensional spaces.

"Principles of Mathematical Analysis" 3rd edition (1974) by Walter Rudin is often the first choice. This book is lovely and elegant, but if you haven't had a couple of Def-Thm-Proof structured courses before, reading Rudin's book may be difficult.

I've recently discovered Lara Alcock's 'How to think about analysis'. It isn't really a textbook, it's more of a study guide on how to go about learning analysis, but I believe it also covers the key ideas.

I read this question a month ago and I decided to go for three of the most suggested books: Abbott "Understanding Analysis", Rudin "Principles of Mathematical Analysis", and Kolmogorov and Fomin "Introductory Real Analysis".

The one I liked most, and I ended up reading entirely, is Rudin's one: I am a PhD student in engineering and I think the level of the book was perfect to me. Two critiques I have are: there is a general lack of comments (a bit too much "Theorem, Proof") and there are no images. However, I found the book very clear and rigorous, especially the first 7 chapters. I definitely suggest it.

I really liked Abbott's approach: he really makes you understand the logic of things, and you never get lost in the proofs. On the other hand the one thing I didn't quite like was the excessive use of exercises: every two pages some kind of proof is "left to the reader." Sometimes also people that are not undergrads are going to read the book! Moreover this book treats only real numbers, and sometimes you lose the "big picture."

If you've had a strong course in Calculus, I highly recommend Advanced Calculus by G.B. Folland. It is well known that Folland's an amazing expositor; this book serves well to introduce you to the crucial transition from Calculus to Real analysis. This book should also prepare you sufficiently in terms of maturity for you to then be able to appreciate Baby Rudin.

The contents are systematically structured with enough attention given to each topic. Some of the topics included in the book are Set Theory, Real numbers, Sets in R, Real Functions, Sequence, Series, Limits, Continuity and Differentiation. The book also contains solved exercises to help the readers understand the basic elements of the topics discussed in the contents

I recommend Courant and John's 'An introduction to Calculus and Analysis', volumes I and II. The authors give a rigorous treatment of their subject while still telling what motivates the ideas. Unlike many modern textbooks, they are not an sequence of definition-lemmas-theorems. These books emphasize ideas over structure. The authors' distinguished careers in applied mathematics ensures that there are plenty of examples and diagrams to illustrate their point.

Those with an 'applied' bent of mind, who want to trace the origin of ideas, not lose touch with the sciences that motivated development of mathematics may find these venerable volumes more rewarding than the modern treatments.

I think a good first book is 'A First Course in Mathematical Analysis' by David Alexandar Brannan and can suggest it as well as several that have already been mentioned on this page, but this one has the advantage that it was a byproduct of the Open University and is thus totally designed for self-study. Lots of problems placed near the relevant discussion, good margin notes for a beginner in analysis, and solutions to check your work.

If you still don't feel ready for Rudin after that, then I can recommend Alan Sultan's 'A Primer on Real Analysis' (which I'd recommend anyways because it should be better known) which is very nice and has lots of pictures to help development of intuition and lots of problems too with most solutions in the back of the book.

Books with so many problems and exercises with their hints and solutions are very appealing. But what you really need is a mature and deep grasp of basics and concepts. After all, that's all you need to tackle these exercises with even a surprising level ease and fun.

I do understand the emphasis on solutions because we all deal with self study, at least sometimes, and solutions/hints are crucial to make an evaluation of your own work.If you are really serious you will soon find out that what you really need are hints not solutions.Needless to say, hints or solutions are supposed to be a last resort, when there seems to be no way out. Even then, a hint is better taken only partially. And by the way: when tackling problems, it is when there seems to be NO WAY OUT that the actual LEARNING process takes place.

I encourage you to take a deep look into The Trillia Group funded, and free, Zakon's books: Mathematical Analysis I which followed by another volume, but to get some basics, Basic Concepts of Mathematics might be a good place to start.In the third mentioned book, this was mentioned:

I did not take these points very seriously, until I started reading and working on it.It is hard to find yourself completely stuck somewhere: it seems that all have been packed for a person who is learning on his own. Hints are provided whenever needed. In many occasions there are questions like "...Why?" which help in following the text rigorously.

I was recommended Introduction to Analysis by Mattuck. It was a bit difficult to use as it does not follow the progression other books (like Rudin or Apostol) follow. Maybe others can share more about their experience with this book, if they have used it.

For ones who read German, I strongly recommend Harro Heuser's 'Lehrbuch der Analysis Teil I'. There is also 'Teil II'. I tried couple of other German text books, but gave up continuing due to many errors or lack of completeness, etc. Then a person recommended me this book.

This book is self-contained and proofs are quite error-free as well as well-written for novices, though personally there were couple of proofs which were difficult to grasp, e.g. Cantor's Uncountability Proof and something else. The author tried to give proofs without the need of studying other subjects of mathematics, e.g. explaining compactness without referring to topology, which sometimes is a hard job. The author revised this book many times (lastest version is 17th edition). I feel sorry that the book has not been updated since the author has passed away in 2011. I recommend reading this book from the top to the bottom, even you have studied with another book before because the author builds up earlier proofs for later ones. I once tried to read from the middle, but gave up and re-started from the top.

I think Ross' Elementary Analysis: The Theory of Calculus is a good introductory text. It's very simple and well explained, but not quite at the level of Rudin's Principles of Mathematical Analysis (for example, everything is done using sequences in Ross, versus a general topological setting for open and closed sets in Rudin). But, if you master it, you can pick up the necessary ancillaries from Rudin or similar pretty quickly. FWIW, Rudin is the standard text for undergrad real analysis.

Another good option is Hoffman's Analysis in Euclidean Space. This was the book MIT used before Rudin arrived, and is a Dover book (so very cheap). I found its exposition to be comparable in level to Rudin, but easier to understand.

Finally, another book I can recommend is Hoffman's Elementary Classical Analysis. This is similar in level to Rudin, but has a lot more material worked out for you. Theres also a tiny bit on applications, so if you're an engineering/science student whose taking real analysis, it can be a bit helpful.

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