# Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

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I don't know if this is the right place to be asking this question. In any case, please explain what "300-400 level" means. I'm guessing that this is US terminology; those outside your country won't necessarily know what it means. (Voting -1 for lack of clarity.) – Tom Leinster Nov 4 '09 at 1:02
I'm in the US, and I don't know what 300-400 level means. I think this is university-specific. – Andy Putman Nov 4 '09 at 1:05
I have known universities where 300-400 means lower-division undergrad, upper-division undergrad, and graduate, and where there are no courses with such numbering. – Theo Johnson-Freyd Nov 4 '09 at 2:03
At the liberal arts school that I've attended introductory real analysis is in the 100s. – Akhil Mathew Nov 4 '09 at 2:28
Questions like this (i.e. asking for a sorted list of questions), must be community-wiki (please see the FAQ). I've hit it with the wiki-hammer. – Scott Morrison Nov 4 '09 at 2:56

Stephen Abbott, Understanding Analysis
Strongly recommended to students who are ony getting to grips with abstraction in mathematics. Find a review here.

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I recommended this book yesterday (it even got 2 votes) but my answer seems to have vanished... – lhf Nov 4 '09 at 11:44
After reading some of this book I have to say I really like it. Many of my complaints about some of the other books that I was looking at were not clear in my head until I saw the same topic in Abbott's book and saw how he explains the purpose of the theorems he presents rather than just giving the theorem and proof. I think for an introductory class students will benefit from that exposition. – Ryan Nov 6 '09 at 23:02
Just to clarify, does this book assume students entering the course already have a basic calculus background? The review seems to indicate that. – Ryan Budney Jan 10 '10 at 19:04
Yes, it does. Not much sense avoiding calculus as a prerequisite. – lhf Jan 11 '10 at 1:43

Anyone that thrusts baby Rudin - as so many departments do, sadly, in an act of either callous indifference or elitist zealotism - on beginning analysis students with no prior experience with rigor is committing an act of inhumanity against a fellow human being. Let's face it: Calculus just ain't what it used to be and Rudin is going to be a buzz-kill for any but the best students. I personally have never liked Rudin even for good students. Rudin seems more interested in showing how clever he is then actually teaching students analysis.
My recommended texts:

• For average students,who have never seen proofs before, I strongly recommend Ross' Elementary Analysis:The Theory Of Calculus.
It's gentle, complete and walks the reader through a careful presentation of calculus containing many steps that are usually omitted or left as an exercise. It can also be used for an honors calculus course: I've had friends that have used it for that purpose with great success. Spivak is a beautiful book at roughly the same level that'll work just as well.
• More advanced, but I think well worth the effort, is Kenneth Hoffman's Analysis In Euclidean Space, which I reviewed for the MAA online a few months ago when Dover reissued it.
It's an amazingly deep and complete text on normed linear spaces rather then metric or topological spaces and focuses on WHY things work in analysis as they do. This is the kind of book EVERYONE can learn something from and now that it's in Dover,there's no reason not to have it.
• Lastly, for honor students on their way to elite PHD programs, we now have a wonderful alternative to Rudin and I'm shocked no one's mentioned it at this thread yet: Charles Chapman Pugh's Real Mathematical Analysis, which developed out of the author's honors analysis courses at Berkeley.
It's terse but written with crystal clarity and with hundreds of well-chosen pictures and hard exercises. Pugh has a real gift that's on display here. He knows exactly how many words it takes to clearly explain a concept-NOT ONE WORD MORE AND NOT ONE WORD LESS. I've never seen any author who does this as effectively as Pugh. The many, many pictures greatly assist him in this task: all of them serve some purpose, none are throwaways just to fill space. Even if it's just to make a joke(see the cornball pic in chapter one showing a Dedekind cut,ugh).

Oh, almost forgot my personal favorite: Steven Krantz's Real Analysis And Foundations. If I was ordered to teach real analysis tomorrow, this is probably the book I'd choose, supplemented with Hoffman. Krantz is one of our foremost teachers and textbook authors and he does a fantastic job here giving the student a slow build-up to Rudin-level and containing many topics not included in most courses, such as wavelets and applications to differential equations. What's most impressive about the book is how it slowly builds in difficulty. The early chapters are gentle, but as the book progresses, the presentation and exercises become steadily more sophisticated. By the last chapter, the presentation is a lot like Rudin's. I would strongly consider this text if I was trying for self study.

Anyhow, those are my picks.

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Rudin covers some important points of point-set topology that are simply not covered in any other introductory analysis book. Students in an "honors calculus" course at the level of math 55 at Harvard (real analysis in disguise) who do not see a fairly significant portion of point-set topology by the end of the first semester are in my opinion being done a huge disservice. – Harry Gindi Mar 10 '10 at 18:56
Your first sentence on Rudin's book is very bad, unfair and very likely not true. Any serious college student who approaches analysis for the first time must know what a proof is, having seen it in Euclidean Geometry back in junior middle school. And I personally like Rudin's, read it when I was still in high school and found it clear, to-the-point, and with a good supply of excellent problems. Also, one cannot fault an author for giving slick proofs. I for one prefer slick proofs over tedious, drawn-out proofs (unless they're the correct conceptual ones). – Anonymous Mar 10 '10 at 19:04
A great deal of point set theory is covered in Pugh and done more clearly then in Rudin. I don't know where Anonymous was trained,but clearly came from a better system then most students come from. Most high schools in America in 2010 have trouble graduating students who can READ,let alone know geometry.It's easy to like slick proofs when you're experienced and well-versed in rigor.Most instructors don't remember what it was like struggling with that fundamental change in thinking that proof creates the first time.Worse,gifted students think anyone that doesn't find it easy is an imbecile. – The Mathemagician Mar 11 '10 at 4:12
@Anonymous High school geometry is VERY poorly taught at most high schools in the US now-if,indeed,it's taught at all. The collapse of the American secondary school system has sadly affected incoming 1st year math students more then any group.We need to adjust the analysis texts accordingly.The students are NOT "dumber" then in previous generations-as a lot of better trained students snark nowadays-they're simply very poorly prepared.@Daniel Take a good,careful look at Pugh's book,especially the exercises. I think you'll find it far superior to Rudin while still remaining terse and concise. – The Mathemagician Aug 14 '10 at 21:11
I haven't read any of the books suggested by @AndrewL so I'm not going to comment on whether they are any good. However, I concur 100% that students entering college now are not as well prepared as those from earlier generations. Rudin is too difficult for a first course simply because they have not been prepared well enough (just look up any recent high school text). It would certainly make a great text for a follow-up course after one has acquired the fundamentals of analysis though. – lightalchemist Sep 17 '13 at 9:31

Look no further than Spivak's completely amazing Calculus. I have taught analysis courses from this book many times and learned many things in the process. One example is the wonderful "peak points" proof of the Bolzano-Weierstrass theorem. The exercises are really good too.

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Rudin's Principles of Mathematical Analysis

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I'm currently taking an introductory course in real analysis at the University of Glasgow. The set text is "Calculus" by Spivak. Totally deserving of its reputation. It's a great read with loads of exercises of varying degrees of difficulty. I also dip into a few others on a regular basis:

• "Calculus", Vols. 1 and 2 by Apostol - a bit drier than Spivak but the exposition is spot on. Great coverage of topics in linear algebra too.
• "A First Course in Mathematical Analysis" by Burkhill - an oldie but a goldie. Surprised it hasn't been mentioned yet.
• "Introduction to Real Analysis" by Bartle and Sherbert - formal, well laid out.
• "Fundamentals of Mathematical Analysis" by Haggarty - a bit more hand holding. A great first text for self study I would say.
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I do not know if it fits in the US curriculum, but to my mind the best book for mathematics undergraduates to learn analysis is Analysis I by Amann/Escher. I used to learn with it in my first 3 semester analysis courses (in Germany). I also quite liked Stephen Abbott's Understanding Analysis, but in hindsight it is not rigorous enough.

Amann/Escher approach each subject from a very general view. For a novice this is most times a bit harder than comparable textbooks, but it pays off. The amazon preview of the English version by Chris Moore sums it up well.

Analysis I is the first in a 3 volume series up to measure theory and Stokes' theorem. It fits quite nicely with the first 3 analysis courses at German/Austrian/Swiss universities.

If nowadays a non-math student comes in my office and is interested in real math, I recommend reading the first chapter of volume I about types of proof and elementary logic to get a first glimpse on how mathematics works.

I can only recommend to have a look at this series.

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I do like Amann/Escher's books and I have based my own analysis courses partially on that series. That said, I find it a "very german" book :), in its style as well as in its level. Very much a top-down-book, whereas my understanding is that current math teaching is going towards bottom-up-approaches. Besides, I think that the non-german world thinks of something completely different when "real analysis" is mentioned: $L^p$-spaces, some elementary functional analysis, perhaps Sobolev spaces, some elementary Hilbert space theory etc. – Delio Mugnolo Oct 21 '15 at 8:09

I was introduced to real analysis by Johnsonbaugh and Pfaffenberger's Foundations of Mathematical Analysis in my third year of undergrad, and I'd definitely recommend it for a course covering the basics of analysis. I'm not sure if it's still in print (that would certainly undermine it as a text!) but even if it isn't, it would make a great recommended resource or supplementary text.

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It's in print as a Dover book now. – Michael Greinecker Nov 21 '11 at 7:52

I'd recommend Hardy's Course of Pure Mathematics. Now in it's 101st year it still remains relevant to modern readers. It takes it bit longer to get to core of real analysis (e.g. limits, continuity, &c., &c.) than perhaps other similar texts do, which tends to make it more suitable as an introductory book, but there's enough there to engage those wanting explore the subjects in more detail.

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I recommend this book: Principles of Mathematical Analysis (by W.Rudin)

By studying this book, you're gonna be able to achieve an accurate, as well as, an abstract view of concepts like continuity or Riemann-Stieltjes Integral ...

By the way, Mathematical Analysis (by Tom M.Apostol) is a FANTASTIC book for one who wants to start the course. I personally taught this book once and the result was great.

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My favourite has always been Introduction to Analysis by Edward Gaughan. I just found out the AMS published the 5th edition. It contains, besides the standard calculus theorems, a very nice introduction to topology of the real line through the study of continuous functions.

I can say that reading this book as a text in my undergrad course largely contributed to myself becoming an analyst.

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There are many good introductions to real analysis. My personal favorite is the UTM by Ken Ross.

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K. Ross, Elementary Analysis: The Theory of Calculus ... Since "UTM" is apparently not on the cover any more, some readers may not know what you meant. – Gerald Edgar Mar 10 '10 at 20:17
@GE: Thanks! And now I'm not sure if it was ever a UTM, or maybe a UTX (or whatever they other series is called). But, yeah, Ross's undergrad yellow book. Part of why I like it is that he's a friend of mine. – Theo Johnson-Freyd Mar 11 '10 at 1:23

I'd recommend Analysis Now. EDIT: Now that the question has been clarified, I'll point out that this is too advanced for a first analysis course.

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Just to clarify that's a graduate/advanced undergrad book, yes? At least one analysis class previously? – Michael Hoffman Nov 4 '09 at 3:20
Yes, it's no good for starters, but it's friendly, well-organised, and a reliable reference. It also has a great title. – Scott Morrison Nov 4 '09 at 4:01
I've been told by at least two different analysts that this is one of those books that every mathematician should have on his shelf. – Ben Linowitz Jan 10 '10 at 18:58
The book has mathematical physics flavor to it, which is not a surprise(author is E. Lieb) I have used it for rearrangement inequalities and some other topics related to my reproach. A very good book.That being said, seems that it is not intended to cover operator theory topics. – BigM Apr 9 at 22:20

I'm using Analysis: With an Introduction to Proof by Steven Lay in my course right now, and from a student's perspective, it's been really good - clear explanations, and a tone of writing that doesn't seem too uptight. I can't speak to other books, but I've enjoyed this one so far!

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I think principle of analysis(rudin) and analysis (tom apostel) is good for you

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JFYI, several posts down you have an answer that is essentially identical to the one you just gave: mathoverflow.net/questions/4023/… for community wiki/ref req type questions, it is often better to vote up the previous posts to show agreement rather than posting anew. – Willie Wong Jan 14 '11 at 19:16

Binmore, Mathematical Analysis. He's at one of the London Universities (UCL I think). It's not flashy but it's very clean. The proofs are there; they're tidy and I think it's readable. I've used it for this kind of course myself.

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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.

Moreover, the book has solutions to all of the exercises.

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

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You might like to try Vladimir Zorich: Mathematical Analysis I. This is an English edition of a Russian textbook, which has been the standard one at Moscow State University. It is comprehensive and very readable.

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I recommend Frank Morgan's Real Analysis for its clarity, the concise chapters, and good exercises. It's much more accessible than Rudin... while I loved learning with Rudin, I don't think it's for everyone.

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I'm not a fan of the Pfaffenberger text. For example, look at the proof of the chain rule. The proof sticks to the "derivative as slope" idea, and so has to consider the special case where one derivative is zero. This isn't very elegant, and causes confusion in what should be a straightforward proof -- IMO when students are first being exposed to something as elementary as analysis, simplicity should be an overriding concern.

Apostol, Buck and Bartle, those are texts that I like pretty well. Or the lecture notes used at the University of Alberta for their honours calculus sequence Math 117, 118, 217, 317 (available on-line) -- pretty well based on Apostol.

There's a few subtle issues going on here. Some departments view analysis as something people learn after they go through a service-level calculus sequence. Some departments treat calculus as part of an analysis sequence -- ie students only see calculus through the eyes of analysis. What book you choose is largely determined by what path your department is comfortable with.

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We use Fundamental Ideas of Analysis from Michael Reed and have been very pleased. It's pretty nice as a 1 semester course for undergrads and has some nice lead ins to other areas where analysis tools are useful.

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Here is a new addition to the literature of books treating Calculus more rigorously than usual:

The How and Why of One Variable Calculus by Amol Sasane, published by Wiley in August 2015.

A google preview can be found at

and some sample material from the book can also be found on the publisher's website for the book, at:

http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119043387.html

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The exact same answer was given here, word-for-word: mathoverflow.net/a/221372/2926 – Todd Trimble Oct 21 '15 at 11:37

Heres' another introductory real analysis book: Introductory Real Analysis, by A.N. kolmogorov, S.V. Fomin, Dover publications.

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A Course in Analysis - Volume I: Introductory Calculus, Analysis of Functions of One Real Variable First of a massive 6volumes set

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