Text for an introductory Real Analysis course. - MathOverflow

Text for an introductory Real Analysis course.

2009-11-04T00:44:34Z

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?

Answer by Steve Huntsman for Text for an introductory Real Analysis course.

2009-11-04T00:55:55Z

Rudin's Principles of Mathematical Analysis

Answer by lhf for Text for an introductory Real Analysis course.

2009-11-04T01:55:09Z

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

Answer by Theo Johnson-Freyd for Text for an introductory Real Analysis course.

2009-11-04T02:05:18Z

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

Answer by Scott Morrison for Text for an introductory Real Analysis course.

2009-11-04T02:56:58Z

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.

Answer by Ross Churchley for Text for an introductory Real Analysis course.

2009-11-04T04:07:01Z

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.

Answer by Ricardo for Text for an introductory Real Analysis course.

2009-11-04T04:38:42Z

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.

Answer by ajm for Text for an introductory Real Analysis course.

2009-11-04T04:48:08Z

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!

Answer by axiomsofchoice for Text for an introductory Real Analysis course.

2009-11-04T09:16:30Z

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.

Answer by Elisha Peterson for Text for an introductory Real Analysis course.

2009-11-04T15:27:19Z

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.

Answer by Ryan Budney for Text for an introductory Real Analysis course.

2009-11-06T19:18:25Z

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.

Answer by Fran Burstall for Text for an introductory Real Analysis course.

2009-11-07T21:33:17Z

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.

Answer by Soheil Malekzadeh for Text for an introductory Real Analysis course.

2010-01-10T17:21:52Z

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.

Answer by Andrew for Text for an introductory Real Analysis course.

2010-01-10T19:56:24Z

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.

Answer by Andrew L for Text for an introductory Real Analysis course.

2010-03-10T18:41:22Z

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.

Answer by Colin Pratt for Text for an introductory Real Analysis course.

2010-03-10T20:41:58Z

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.

Answer by vahid for Text for an introductory Real Analysis course.

2011-01-14T19:12:43Z

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

Answer by Anthony Quas for Text for an introductory Real Analysis course.

2011-01-14T19:49:56Z

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.