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Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results that would serve as a very good reference book for specialist analysts in any field, whether functional, complex and measure theorists. Like change of limits, convergence of series etc.

I notice the question on undergraduate textbooks has few responses regarding analysis books of this sort.

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Um, is there any reason why you're against Rudin, Spivak, and Stein-Shakarchi? Those were the 3 that I would recommend! –  Harry Gindi Dec 31 '09 at 21:05
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Your requirements seem somewhat contradictory: an undergraduate text is an intended as an introduction accessible to the widest possible audience. I don't know of an undergraduate text in any subject that I would describe as being comprehensive, containing general results, and being a very good reference book for specialists in the field. In fact I think Rudin's book is about the best you'll find in this regard. –  Pete L. Clark Dec 31 '09 at 22:49
    
Assuming we mean the same thing by "undergraduate", of course. I am taking the term in the North American sense, which implies someone who has not yet devoted the entirety (often, not even the majority) of their academic studies to mathematics. –  Pete L. Clark Dec 31 '09 at 22:51
    
But I do agree that Spivak and "little" Rudin are both undergraduate texts. Stein-Shakarchi seems on the borderline. (Sorry for all the comments.) –  Pete L. Clark Dec 31 '09 at 22:53
    
I disagree that "undergraduate" means what you've said. I would take undergraduate to mean "doesn't assume prior knowledge in the specific subject" and "doesn't require very much mathematical maturity". Books like Artin's Algebra, Hoffman and Kunze, or Baby Rudin fit this bill perfectly. –  Harry Gindi Jan 1 '10 at 1:23
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29 Answers

up vote 7 down vote accepted

Nobody has mentioned Folland's "Real Analysis with Applications"?? This was the textbook for my undergraduate real analysis course (measure theory, Banach spaces, Hilbert spaces), and I still go back to it all the time. I am not yet all that experienced (I just finished my third year of graduate school), but overall I have gotten more use out of this book than any other that I own.

It has the most comprehensive swath of applications of analysis of any introductory text I have ever encountered: basic functional analysis, Fourier theory, probability theory, distributions, Hausdorff measures, Haar measure, smooth measures, and more. The early material is covered with all the appropriate detail, while the later material quickly provides the essential definitions and results needed to come to grips with an unfamiliar idea in the literature. Also, the exercises are abundant and uniformly fantastic. My only complaints are that some of the later proofs are hard to read, and there is sadly no discussion of the spectral theorem.

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While I'm a huge fan of Folland's book, I wouldn't call it an undergraduate text... –  Mark Jun 28 '11 at 11:15
    
@Mark Totally agree.Undergraduate at Harvard or in Germany,MAYBE. But not at universities for mere mortals. –  Andrew L Jul 8 '11 at 16:18
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I and perhaps a dozen of my undergraduate colleagues took a class based on this book during my third year at the University of Michigan. And I can assure you, I am quite mortal. –  Paul Siegel Jul 14 '11 at 12:53
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@Andrew: I can also ensure you that most Germans are mortal. Although I assume that no current student of a German university is dead, so they might be immortal... –  Lennart Meier May 5 '13 at 18:33
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Charles Pugh's Real Mathematical Analysis covers a wide range, starting from real numbers, topology, and basic 1D calculus, and then moving into multivariable calculus, function spaces, and Lebesgue measure/integration, all in a compact 450 pages. The writing is clear and quirky, and there are lots of interesting and hard problems.

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I found it a bit too quirky. –  Kevin H. Lin Jan 4 '10 at 1:07
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I love Pugh's book,but I think it needs a long preface talking about it's history,philosophy,etc. I think this would greatly enhance the book and give people a better idea of it's intended audience and Pugh's ideas of how to teach analysis.You'd think someone who taught this course for 30 years at Berkeley would have a LOT to say on the subject. –  Andrew L Mar 25 '10 at 4:22
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T.W. Korner has a book (there is a draft here). On the off-the-beaten-track side, you can always use Keisler. Finally, for some good-old dialectic materialism and entertainment value you can use G.M. Fichtenholz (couldn't find a link to that - but they still used a Hebrew translation (well - more or less) in Israel not so long ago).

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Isn't "dialectic materialism" a term used by old Marxists? =p –  Harry Gindi Jan 1 '10 at 1:44
    
@Harry: Indeed it is - it's the only calculus book I've seen which quotes V. I. Lenin - as I said dialectic materialism and entertainment. –  David Lehavi Jan 1 '10 at 6:42
    
Your link for Kiesler is the same as your link for the Korner book. And did you mean Jerome Keisler, with an "ei"? –  Ben Crowell Sep 28 '12 at 22:09
    
@Ben - oops. fixed both. Thanks –  David Lehavi Sep 30 '12 at 3:30
    
@DavidLehavi: The Keisler link is broken. –  5space May 29 at 6:01
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Terence Tao has published his notes for undergrad analysis as a book:

http://terrytao.wordpress.com/books/

The original notes can be found on his webpage. I'm not sure exactly what the differences between the notes and the book are.

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These notes are awesome and would love to use them and Pugh for an honors course one day! –  Andrew L Jul 6 '10 at 7:38
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I liked Kolmogorov & Fomin's books when I was an undergrad. Not much complex analysis in them, but they're great if you like functional analysis.

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I only know ONE Kolmogorov & Fomin book, "Elements of theory of functions and functional analysis". We used it for functional analysis, but does it cover anything else? –  Victor Protsak May 24 '10 at 5:32
    
@Victor: there is (sort of) another book by these authors called Introductory Real Analysis. This is billed as a translation of a Russian book by these authors -- which I presume is the book you're thinking of -- but the translator, Richard A. Silverman, really did more than just translate, as he explains in the introduction. Anyway, this was one of the texts for my undergraduate real analysis course, and I remember it reasonably fondly. (Yes, there is more to it than functional analysis per se.) –  Pete L. Clark Mar 24 '11 at 9:41
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I learned real analysis from Strichartz' wonderfully titled book, The Way of Analysis. It is very wordy, but I really liked it. I suspect that I would like it less as an instructor, simply because in preparing a lecture I don't need all the discussion, I just want to remember the main point. But for a student, the discussion in this book can be quite beneficial, and of course as instructor, one can always keep a copy of Rudin close by.

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I liked it quite a bit,too,Dan. I think it's one of the great unsung analysis texts,especially for students brave enough to try and learn real analysis on thier own. –  Andrew L May 24 '10 at 5:00
    
I've taught "Honors real analysis" from Strichartz a few times and really liked it. We also used to place Rudin on course reserve. In a curious role reversal, one of my students wrote "this book is chatty, you should have used Rudin instead". One cautionary remark: some of the exercises are way off and a few 2-line proofs have an extra page to them (I asked Bob what he meant there and he confessed that he no longer remembered; a continued fraction exercise from an early chapter seemed disconnected from the text, but the solution he gave me, frankly, I wouldn't have accepted from a student.) –  Victor Protsak May 24 '10 at 5:26
    
(I have deleted Andrew L's reply.) –  S. Carnahan Aug 11 '10 at 14:43
    
One really trivial point that drives me crazy about Strichartz: he doesn't know how to spell Schwarz (as in Cauchy-Schwarz). Or maybe he just can't overcome the force of habit from the spelling of his own name. –  Mark Meckes Aug 19 '11 at 17:48
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The Student Mathematical Library has three volumes of analysis problems for undergraduates:

The best part of analysis at this level is how it enhances calculus.

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Roger Godement Analysis(I-IV-in french,I-II-in english)contains more than Bourbaki's "Functions of one real variable",has motivation and historical insight(not quite a textbook however..)

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V. A. Zorich's Mathematical Analysis I and II (Springer). It covers undergraduate material from an advanced viewpoint, contains lots of good physically oriented examples, and is quite comprehensive.

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VERY good choice,Slobadan-and a book that's strangely little known on this side of the Atlantic. –  Andrew L May 24 '10 at 4:59
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Also Abbott's Understanding Analysis.

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Don't miss the MAA Review, which starts like this: "This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks" –  lhf Jan 25 '10 at 0:29
    
I have taught from Abbott's Understanding Analysis, and it worked well. However, I do think it is slightly "dangerous" in the sense that Abbott uses the sequential definition of continuity in place of epsilon-delta, without any warning that the equivalence of these two definitions depends on the axiom of choice. –  John Stillwell Mar 3 '10 at 22:07
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I've just taught from Abbott and I was very annoyed by the overly casual style. I don't mean freshness of the exposition, that's wonderful, just the sad fact lots of things are either left to the reader to figure out or are explained carelessly (typical example: the proof of existence of the square root of 2 in Chapter 1). For inexperienced students, it's a disaster, and regardless of the level, if one cannot assign the proofs to be read in the text, what use is the text? While I love Cantor set as a unifying theme in a real analysis course, nearly everything else about this book is negative. –  Victor Protsak Jun 29 '11 at 5:42
    
I agree with Victor, having taught from Abbott in 2002 and 2003. The biggest drawback is the discussion of derivatives without any mention of tangents. –  Margaret Friedland Sep 27 '12 at 20:31
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Karl Stromberg
Introduction to Classical Real Analysis (Wadsworth & Brooks/Cole Mathematics Series)

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Knapp's "Basic Real Analysis" covers a lot of material and takes care with some of the topics you mentioned. I'm not completely sure if I would have wanted it as my very first analysis book but it would have been good to have at hand and I think it would be a good text to work through.

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The recently released book Real and Complex Analysis, by Apelian and Surace, covers basic real and complex analysis together at an undergraduate level.

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Minor disclosure: I was involved in its production. –  Akhil Mathew Dec 31 '09 at 15:54
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Zygmund and Wheeden's Measure and Integral is quite nice, and compact. We used in at the University of Alberta for our 4th year analysis sequence (a full year course). It does assume the students coming into the course are fairly mature in how they think about mathematical formalism. Probably not the right textbook for a group of students coming out of a purely "service" calculus sequence.

On the other end, Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach is sort of like a souped-up service-calculus course that's bordering on being analysis. You don't go so far as measure theory, but you do calculus so well some people might consider it to be a baby analysis course.

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I was living in the guest room in Hubbard's attic for one week, and he was writing the latter parts of Chapter 6 of his book, which I got to proof read while I was there. I already knew the material, but the clear explanation helped me learn new things about the topic. –  Jacques Carette Mar 4 '10 at 1:51
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We used Real Analysis by N.L. Carothers when I took my first course on metric spaces quite some years ago. Its only weakness is that it doesn't have any material on Hilbert spaces (at least if I recall correctly) and that you probably need the students to have learned of real sequences, real continuous functions and the Riemann/Darboux integral in an earlier course.

Link: Amazon, Google Books.

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I find this question strange because Stein-Shakarachi series covers different things compared to baby Rudin. Anyway if you are looking for some notes that cover basic analysis, here is one. It covers from basic calculus up to multivariable integration (Jordan content), and along the way introduced topology on $\mathbb{R}^n$ and differential forms on $\mathbb{R}^n$. Originally there was also a brief introduction to measure theory (up to Radon-Nikodym), but it seems that this part was put in another file.

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Amann, Analysis I, II, III

Both original German version, and English translation.

It contains most important things 'all-in-one', really comfortable to read.

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The first analysis course I ever took used the book "Elementary Analysis" by Ross. It's basically baby-baby-Rudin. Ross's book (if you include the exercises and the optional sections) covers more or less the same material as the first 8 chapters of baby-Rudin, but the exposition is much friendlier and it's more easy-going for a beginner. When I say "beginner" here I really mean beginner -- someone who has never even written a rigorous mathematical proof. The book would probably be very boring and tedious for someone above this level.

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This was the first of the baby real variables texts and it's still a classic.I can't think of better summer reading for an honors high school graduate in mathematics about to enter MIT. –  Andrew L May 24 '10 at 5:02
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Gaughan's book and the book by Swartz and Depree are excellent for undergraduate Analysis. Swartz Depree also does the Gauge integral.

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I'm surprised no one has said Marsden and Hoffman's "Elementary Classical Analysis", but perhaps it is too elementary or classical. I didn't learn from it as an undergrad, but I did find myself turning to it as I worked problems from "Berkeley's Problems in Mathematics" by de Souza and Silva. M&H fleshes out a lot more detail, which Rudin spares for the sake of elegance or relegates to the exercises. I wish I had it or Korner's book "A First Second or Second First Course in Analysis" alongside Rudin when I first studied analysis. In particular, I prefer Marsden and Hoffman's treatment of Arzela-Ascoli over Rudin's.

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I find Marsden and Hoffman a bit hard to read, because of the strange device of separating the proofs from the theorems. –  Michael Lugo Jan 7 '10 at 15:33
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I learnt measure theory and some other analysis from Royden: "Real Analysis". That was quite interesting!

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I just took my first analysis course and used "Fundamental Ideas of Analysis" by Michael Reed and enjoyed it very much. Good motivating points, good problems, and has allowed me to access other, more deep, analysis texts.

link: Amazon

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Shilov's Elementary Real and Complex Analysis is comprehensive, straightforward, and as a Dover book, is excellently priced.

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There's Yeh's Theory of Measure and Integration which covers almost everything Folland does, but in a (really) verbose fashion.

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When I was studying real analysis we used Folland, and when some part got rough or I got lost in technicalities, I would seek it up in Yeh and be very satisfied. –  Bruno Stonek Jul 7 '11 at 21:46
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A very alternative approach is Carol Schumacher's Closer and Closer: Introducing Real Analysis which uses inquiry-based learning / the Moore method. (http://www.jblearning.com/catalog/9780763735937/)

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There's a period of time when I'm trying to substitue the whole undergraduate real analysis course with the alternative choice: Use a general topology book like Klaus Janich's Topology and a decent measure theory book like Cohn's Measure Theory. The material I mentioned above can be a ponderable substitution of a real analysis course if you really hate the classics so much...

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Jänich isn't focused enough and requires to much knowledge of other areas to be used to its potential. –  Michael Greinecker Jan 31 at 0:59
    
Agreed. But it's too time consuming to use Munkres. I also recommend the Topological Analysis by Whyburn. –  Henr.L Jan 31 at 3:50
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A really great new book out is "Real Analysis for the Undergraduate - With an Invitation to Functional Analysis" by Matthew Pons. This book offers a classical treatment of undergraduate analysis, but then offers a nice introduction to measure theory and integration theory. Also, at the end of each chapter there are really nice glimpses into ideas from functional analysis and operator theory.

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