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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 2009 at 1:02
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 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 2009 at 1:05
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 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 2009 at 2:03
At the liberal arts school that I've attended introductory real analysis is in the 100s. – Akhil Mathew Nov 4 2009 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 2009 at 2:56
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17 Answers

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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 2009 at 11:44
@lhf: Your answer didn't vanish. I edited it to add links, and answer the second part of the question. – Sonia Balagopalan Nov 4 2009 at 14:22
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 2009 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 2010 at 19:04
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 Yes, it does. Not much sense avoiding calculus as a prerequisite. – lhf Jan 11 2010 at 1:43
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Rudin's Principles of Mathematical Analysis

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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 2010 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 2010 at 1:23
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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 2009 at 3:20
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 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 2009 at 4:01
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 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 2010 at 18:58
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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 2011 at 7:52
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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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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'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 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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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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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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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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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 thier 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 buildup 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 2010 at 18:56
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 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 2010 at 19:04
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 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. – Andrew L Mar 11 2010 at 4:12
I have to say, that Rudin was the best thing that happened to me. While it may not be "user-friendly," it forced me to actually understand the concepts, instead of just parroting the proofs the prof did in class. – Daniel Miller Aug 14 2010 at 12:20
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 @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. – Andrew L Aug 14 2010 at 21:11
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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 think principle of analysis(rudin) and analysis (tom apostel) is good for you

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2 
 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 2011 at 19:16
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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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