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Does anyone 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.

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Related: Good Textbooks for Real Analysis and Topology – MJD May 22 '14 at 13:07

20 Answers 20

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).

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I used Abbott as my introductory text for a first semester of analysis, and it is a really good introduction. Highly recommended. At the same time, it is only an introduction! – nayrb Aug 12 '13 at 15:46
Note that there aren't answers for Abbott – fread2281 Feb 10 at 17:55

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.

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I like Strichartz's book a lot for this purpose. To me,the less formal instruction someone's getting for a subject,the MORE detailed the book should be because then the book IS the teacher. For coursework, just the opposite should be true. Self-study to me is different from asking for a text for coursework in this way. – Mathemagician1234 Sep 6 '11 at 17:08

"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.

Thomas's calculus also seems to fit well to your needs, as i myself had used that book and found it more appealing than Rudin's

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Rudin, as a beginner's text for self-study? Really? – ItsNotObvious Sep 6 '11 at 15:59
No comment.......... – Mathemagician1234 Sep 6 '11 at 17:09
I've had transition to advanced math where you learn to write proofs. I was a math major before dropping out but it's been about 10 years since I last looked at this material. CritChamp – CritChamp Sep 6 '11 at 20:09
No doubt that baby Rudin is great to read. But for a beginner's level, it looks too abstract and too many intuitions and details are either omitted or over-simplified. Nonetheless, for those who wish to truly understand real analysis, baby Rudin is definitely worth their effort, even for beginners! – Vim Feb 24 at 3:12

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.

[1] Victor Bryant, "Yet Another Introduction to Analysis", Cambridge University Press, 1990.

[2] David M. Bressoud, "A Radical Approach to Real Analysis", 2nd edition, Mathematical Association of America, 2006.

[3] Stephen Abbott, "Understanding Analysis", Springer-Verlag, 2001.

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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.

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I think I will like this book very much. Thanks! :) – Srivatsan Sep 6 '11 at 16:27
3Sphere: you need to include the http:// in hyperlinks. Also, you should include them as "http://...", not as 'http://...'. – t.b. Sep 6 '11 at 18:22
@Theo Thanks for the tip. – ItsNotObvious Sep 6 '11 at 18:55
What are o-O concepts? – Rudy the Reindeer Nov 14 '12 at 9:01
Also, the link included in your answer doesn't work anymore. A Google search came up with this, although I don't know whether this is the latest version. – Rudy the Reindeer Nov 14 '12 at 9:04

I like Terrence Tao's Analysis Volume I and II. By his simple way of explaining things, this book must be readable by yourself.

You can see here all his books along with the two, I mentioned above.

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Mathematical Analysis I & II by Vladimir A Zorich, Universitext - Springer.

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Why do you recommend it? – Antonio Vargas Dec 6 '14 at 6:34
It has good number of examples and the explanations are lucid. – Sourav Sarkar Sep 15 '15 at 22:15

I recommend Mathematical Analysis by S. C. Malik, Savita Arora for studying real analysis. A very detailed and student friendly book!

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The book of Bartle is more systematic; much clear arguments in all theorems; nice examples-always to keep in studying analysis.

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The OP may be unfamiliar with textbooks on this topic. It would be better if you could state the full title of the book rather than just a family name. A google search, e.g., reveals that there is an "Introduction to Real Analysis" by Bartle and Sherbert and also a book called "The Elements of Real Analysis" written by Bartle, and I have no idea which book (or even something else) you are talking about. – user1551 Sep 6 '11 at 7:24
And, let's not forget The Elements of Integration and Lebesgue Measure, also by Bartle! – ItsNotObvious Sep 6 '11 at 16:32
I taught from The Elements of Real Analysis several times a number of years ago; it’s quite good, though the last time I taught the course I substituted my own notes on the gauge integral for his treatment of the Riemann-Stieltjes integral. Bartle & Sherbert is a watered-down version and in my opinion not nearly so good. – Brian M. Scott Nov 14 '12 at 5:54

I really like Fundamental Ideas of Analysis by Reed. It's a friendly and clear introduction to analysis.

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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.

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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.

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See the book S.C.Malik Savita Arora "Mathematical Analysis".

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If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. – martini Nov 14 '12 at 6:37

Might not be a textbook but a very good supplement to a textbook would be the following book Yet Another Introduction to Analysis by Victor Bryant.

As a prerequisite the book assumes knowledge of basic calculus and no more.

This book may be a better starting point for some people.

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1) Introduction to Real Analysis by mapa-

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

2) Elements of Real Analysis by denlinger

Two best books for self-study. Rudin and bartle are good if you have an instructor or in college but for self understanding these are best.

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I found Real analysis by Frank Morgan published by AMS a very nice introduction and Methods of Real analysis by Richard Goldberg a next one.

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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.

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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.

Volume I focuses on calculus on the real line while volume II teaches functions of several variables. On their way, they teach exterior differential forms, ODE, PDE and elementary complex analysis.

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.

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I would recommend "Guide to Analysis" by Hart & Towers which is aimed at those making the transition from high school mathematics to university mathematics and university analysis in particular. This seems like the most sensible choice.

However, the classic text to study real analysis would be "Principles of Mathematical Analysis" by Rudin. If you have not studied much mathematics before it may be tough going.

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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.

The book also has good number of excercises and hints/solutions to selected problems at the end of the book, which I found good for self-learning.

This book assumes no prerequisites, but learnig other subjects parallely is always a good thing with math because it is hard to completely isolate a math subject from others.

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