# Looking for a rigorous analysis book

I'm a mathematics undergrad student who finished his first university year succesfully. I got courses of calculus, but these weren't very rigorous. I did learn about stuff like epsilon and delta proofs but we never made exercises on those things. The theory I saw contained proofs but the main goal of the course was to succesfully learn to solve integrals (line integrals, surface integrals, double integrals, volume integrals, ...), solve differential equations, etc.

I already took proof based courses like linear algebra and group theory, so I think I am ready to start to learn rigorous real analysis, so I'm looking for a book that suits me.

I want the book to contain the following topics:

The usual analysis stuff:

• a construction of $\mathbb{R}$ or a system that takes $\mathbb{R}$ axiomatically for granted
• rigorous treatment of limits, sequences, derivatives, series, integrals
• the book can be about single variable analysis, but this is no requirement
• exercises to practice (I want certainly be able to prove things using epsilon and delta definitions after reading and working through the book)

Other requirements:

• The book must be suited for self study (I have 3 months until the next school year starts, and I want to be able to prepare for the analysis courses).

I have heard about the books 'Real numbers and real analysis' by Ethan D. Block and 'Principles of mathematical analysis' by Walter Rudin, and those seem to be good books.

• How easy did you find those courses? – user49640 Jun 21 '17 at 13:29
• I think on balance I'd probably recommend Apostol's book. I read through about 3/4 of it and found it great. It's less concise than Rudin, but it's more pleasant and has a greater variety of interesting material. It's also good to have an introduction to Lebesgue integration at this level. The multivariable stuff is much better than in Rudin. The main drawback is no manifolds and differential forms. Also, both Rudin and Apostol share another problem, which is that they do Riemann-Stieltjes integration directly, instead of Riemann, which is easier. – user49640 Jun 21 '17 at 13:46
• Take stuff from Zeilberger and Wildberger with a grain of salt since neither of their views are widely held and both have a tendency for inflammatory rhetoric which sometimes leads to misunderstands (to put it judiciously) although this is mostly Wildberger. – lordoftheshadows Jun 21 '17 at 19:01
• I am a fan of Rosenlicht's analysis book (the beige one from Dover) which does everything in multiple variables. I took a class on Bartle and Sherbert. In retrospect either of these seem like a gentle but complete and concise treatment. I've heard the Rudin's analysis books are obnoxiously, unnecessarily difficult to self-study from. I am convinced they are mostly name dropped for intimidation and bragging rights. – Christian Chapman Jun 22 '17 at 15:03
• Possible duplicate of Good book for self study of a First Course in Real Analysis – Christian Chapman Jun 22 '17 at 15:04

Spivak's Calculus is still the best book for a rigorous foundation of Calculus and introduction to Mathematical Analysis. It includes, in its last chapter, very interesting topics, such as construction of transcendental number and the proof that e is transcendental, and the proof that $\pi$ is irrational. It also includes, in the Appendix, a rigorous construction of the set of real numbers by Dedekind cuts.

It is, in my opinion, by far the best Calculus book, if one wants to understand well the $\delta-\varepsilon$ definitions, and be able to solve challenging problems, which require these definitions. One of my favourite Spivak problems of this kind is the following:

Let $f:\mathbb R\to\mathbb R$ be a function $($not necessarily continuous$)$, which has a real limit at every point. Set $$g(x)=\lim_{y\to x}f(y),\quad x\in\mathbb R.$$
Show that $g$ is continuous.

However, Spivak's book treats only one-dimensional Calculus.

Second reading, right after Spivak: Principles of Mathematical Analysis, by W. Rudin. Apart from a good introduction of the Metric Space Theory (to learn what is open, closed, compact, perfect and connected set), there is a number of results on convergence of sequences of functions, multivariate calculus, introduction of $k-$forms and introduction to Lebesgue measure.

As a sequel, one should consider the great little classic, Spivak's Calculus on Manifolds, which provides an elegant and concise introduction of $k-$forms and proof of Stokes Theorem in Euclidean spaces and manifolds.

• Don't forget the sequel, Calculus on Manifolds, which covers the middle ground -- analysis on R^n specifically rather than a general metric space. I highly recommend the second book to anyone who has finished the first book or any equivalent introductory analysis book. – AlexanderJ93 Jun 22 '17 at 20:46
• @AlexanderJ93 Good point! I should include this in my answer. – Yiorgos S. Smyrlis Jun 22 '17 at 20:59

Rudin's text is good and has almost everything you want. But I feel that Rudin + some other book may suit your purposes better.

• Terence Tao's Analysis- 1 describes construction of $\mathbb{R}$ very well. Read the first answer to a question that I asked a while ago here--Good First Course in real analysis book for self study
• Rudin has rigorous development of limits, continuity, etc.. but so do Bartle, Sherbert's Introduction to real analysis and Thomas Bruckner's Elementary real analysis. The latter two deal with single variable only and contain really elementary examples of proving limits, continuity using $\epsilon-\delta$ definition, I don't remember Rudin's text having such solved examples. Worth checking out in my opinion.
• Rudin no doubt has very good exercises and if you get stuck at any of them, there are solutions available online in a pdf and very helpful companion notes-here for understanding theory better with an exercise set at the end of each chapter preparing you for Rudin's exercises.
Though reading it can be sometimes be very frustrating what with lack of examples. I usually advise people to first read through a gentler text like Sherbert then come back to it.
Browse through all the books first and if you feel you're ready for Rudin's, go for it.
• After you're done with whatever analysis text you choose to read, this three problem book set published by AMS is very good. Read more about it here- Problems in Mathematical Analysis.
• Other good books I've heard of but personally have no experience in-Serge Lang's undergraduate analysis, Charles Pugh's Real Mathematical Analysis, Stephen Abbott's Understanding Analysis.
• The link mentioned at the third bullet doesn't work. – user170039 Jun 21 '17 at 13:46
• That's strange. I can't get it to work. Just google "Rudin companion notes" the link I've given here is the same as the first ucdavis link. I'd be thankful if someone could edit it to make it work. – Shreya Jun 21 '17 at 13:55
• I'm going to second the recommendation for Pugh's book. It's roughly at the same level as Rudin but significantly more readable and it's the book I used for self study when I was in a similar situation to OP. – lordoftheshadows Jun 21 '17 at 14:12
• @shrey get rid of the final "/" in the link to fix it – Carmeister Jun 23 '17 at 0:53
• Thanks Carmeister. LInk is fixed now. – Shreya Jun 23 '17 at 18:58

The reason why I shall never write a Calculus textbook is because Michael Spivak's Calculus is a masterpiece written at a level that I would never be able to attain.

If you find it too advanced, I suggest that you read first another book by Spivak: The Hitchhiker's Guide to Calculus.

• I disagree with this recommendation. Spivak is good either for someone who is learning calculus for the first time or for someone who is not yet proficient at proofs and needs time to transition to analysis. For the OP, who already knows the computational aspects of single- and multi-variable calculus, this would be too repetitive. I think the suggestions made by others to use Rudin, Apostol or Zorich would be better in the OP's circumstances. – user49640 Jun 21 '17 at 12:59
• @user49640 I tried Spivak with little Calculus background and I got murdered. Spivak is terrible for a beginner. (For someone having computational experience, though, it should be nice) – étale-cohomology Jun 22 '17 at 17:16
• @étale-cohomology People's experiences with Spivak vary according to their pre-calculus preparation and their ability to adapt to rigorous mathematics. Spivak writes that the book is intended to make calculus "the first real encounter with mathematics." Not only has the OP already mastered the mechanical aspects of calculus, he or she has also been successful in learning group theory. For somebody with that level of preparation, I don't see that there is much to be gained by reading Spivak instead of an analysis text. – user49640 Jun 22 '17 at 17:23
• @étale-cohomology Please, don't make such strong claims as "terrible for a beginner". It is my favourite freshman book and it was a bridge for me towards the rigorous math. I hope your comment won't scare people who can profit from this book. – Mihail Jul 10 '17 at 15:58
• @Mihail It sure scared me! – étale-cohomology Jul 11 '17 at 13:07

Vladimir A. Zorich Mathematical Analysis I and II.

• I agree that Zorich could be a good choice. It's noteworthy that Volume II discusses topological spaces, differential calculus in Banach spaces, and the exterior calculus on manifolds, which are not done or not done well in Apostol and Rudin. These are important topics everyone must study at some point, and it makes sense to have them in a basic analysis course. The table of contents can be found here. Volume 1 does cover some calculus topics, but in a sophisticated way. – user49640 Jun 21 '17 at 13:07

I'm going to highly recommend Pugh's Real Mathematical Analysis. I used it for my first introduction to rigorous analysis and quite liked it. In particular I think it is a good alternative to Rudin since it treats analysis at a similar level of rigor in a much more readable manner.

It has an excellent introduction to Real Analysis in a single variable and a good (but not the best) introduction to multivariable analysis. In particular his treatment of topology is much nicer than is in Rudin and there are an enormous number of problems of all difficulty levels (1 sentence proofs to former Putnam problems).

A word of warning, his style is a bit quirky which I know some people don't like. For me this was a plus but it's not for everyone.

If Pugh/Rudin are too fast for you then I also reccomend Abbott's Understanding Analysis for a very well written introduction that takes things slower and fills in the details more than Rudin/Pugh.

I think Apostol's Mathematical Analysis is pretty good for what you're describing, but you should see here: Rudin or Apostol for a discussion of the merits and demerits of it.

• That amount of mathematical maturity can be obtained reading another book written by Spivak: The Hitchhiker's Guide to Calculus. – José Carlos Santos Jun 21 '17 at 11:15
• @JoséCarlosSantos is that worth mentioning in your answer, maybe? – postmortes Jun 21 '17 at 11:17
• Thanks for your suggestion. I just did that. – José Carlos Santos Jun 21 '17 at 11:21
• It seems to me that Rudin assumes much more mathematical maturity than Spivak's Calculus does. In fact, I believe Spivak's Calculus tries hard to help develop that maturity in the reader. (Are you sure you weren't thinking of Spivak's other book Calculus on Manifolds?) – littleO Jun 21 '17 at 20:08
• @littleO Oh dear, I have some more books to buy then. Don't get me wrong, I loved CoM and it was at a good level for me, but I think I'd benefit from something a step back too. – detly Jun 22 '17 at 3:50

Shrey mentioned it in the bottom of his answer, but I can vouch for Stephen Abbott's Understanding Analysis, followed by Rudin.

Background: I read thru and did all practice problems for Understanding Analysis in about 2-3 weeks and then tackling the beast aptly named Baby Rudin wasn't so difficult. It's a nice mix between conversational and rigor, that serves as a good intro-intermediate level book. It's also relatively cheap if you order it from Amazon. The only con I can think of is that solutions are not provided to the practice problems, which is not that big a deal with M.SE. No matter your skill level I would certainly recommend this book as a good intro into Analysis.

I am surprised no one has mentioned A course of pure mathematics by G. H. Hardy. That book is, in my opinion, a piece of art. It is considered a classic on this topic and has all the features you're asking for, and much more. There are many reviews of this book online, including this Wikipedia article, so I won't write a new one.

• I think the fate of that book is precisely because of the fact the book is too good. After all we have the saying: no good deed goes unpunished. Those books by Rudin and Spivak are nowhere near this classic. Another key aspect is that this book does away with the need for teachers/instructors and perhaps therefore never a favorite among teachers. – Paramanand Singh Jul 3 '17 at 5:50

The Way of Analysis by Strichartz was my undergraduate text. This book is long on explanation and very good at providing intuition. As such, it is unusually well-suited for self-study.

Do not be surprised or ashamed if you are unable to slog through reference texts such as Rudin on your own. They are unsuitable for self-study by most people.

• Rudin was so hard for me the first time, even with solid background. I can't believe I had to scroll all the way to the bottom to find this answer. Strichartz helped me to learn analysis and Rudin helped me to master it. – Alfred Yerger Jun 23 '17 at 7:27

I encurage you to study on baby Rudin, it could be laconic and dry but is rigorous and complete. It contains a construction of real numbers from rational numbers through Dedekind cuts (Appendix 1.8). Chapters 2 contains elements of topology in metric spaces (concept as compactness which is foundamental in analysis). In Chapters 3, 4, 5, and 6 there are limits, sequences, cauchy sequences, series, continuity, derivation, theory of integration. At the end of each chapter there are a lot of challenging problems. I've integrated rudin's study with Apostol books (Calculus vol. 1 and 2) and Francis Su lectures on analysis (https://www.youtube.com/playlist?list=PL0E754696F72137EC).

• I think the main weakness of Rudin's book is the sketchy treatment of multivariable differential calculus, multiple integrals (especially), and Stokes' theorem. Apostol's analysis book also just has a lot more fun theorems in it (but no Stokes, in the second edition). – user49640 Jun 21 '17 at 13:16

Since you're planning to actually take analysis courses in a few months, rather getting one of the standard real analysis texts that others have suggested, I recommend looking at Andrew M. Gleason's Fundamentals of Abstract Analysis.

Here are some comments I wrote about Gleason's book in this 3 January 2001 sci.math post:

I read bits and pieces of the 1966 edition throughout my undergraduate years. This book is VERY carefully written and EVERYTHING is developed from scratch. From what I recall, the book begins with truth tables and propositional logic, then it proceeds to predicate logic, then to set theory, then to the Peano axioms for the natural numbers and a model of them in ZF set theory, then to constructions of the integers, rational numbers, real numbers, and complex numbers, ... Gleason gives a lot of carefully written explanations but somehow still manages to get all the way up to things like the Cauchy integral formula.

The following is from the Wikipedia article on Andrew M. Gleason, quoted from "a reviewer" of Gleason's book:

This is a most unusual book ... Every working mathematician of course knows the difference between a lifeless chain of formalized propositions and the "feeling" one has (or tries to get) of a mathematical theory, and will probably agree that helping the student to reach that "inside" view is the ultimate goal of mathematical education; but he will usually give up any attempt at successfully doing this except through oral teaching. The originality of the author is that he has tried to attain that goal in a textbook, and in the reviewer's opinion, he has succeeded remarkably well in this all but impossible task. Most readers will probably be delighted (as the reviewer has been) to find, page after page, painstaking discussions and explanations of standard mathematical and logical procedures, always written in the most felicitous style, which spares no effort to achieve the utmost clarity without falling into the vulgarity which so often mars such attempts.

• I'm looking at the 1966 edition, and I can't find anything about the Cauchy integral formula (or about integrals, in fact). Can you tell me where that is? – user49640 Jun 21 '17 at 15:34
• I haven't looked at the book in many years, but I believe there is a chapter on complex functions (complex variables, analytic functions, etc.) that is the last chapter of the book, or perhaps the next to last chapter of the book. (moments later) I just looked at the amazon.com table of contents preview, and it's chapter 15 titled Introduction to Analytic Functions. It's possible I was wrong about the Cauchy integral formula, and perhaps only power series results are discussed (I can't tell from the section titles), but nonetheless I do vaguely recall it being in the book. – Dave L. Renfro Jun 21 '17 at 15:55
• I tried some word and phrase searches here (use the "From inside the book" window), and now I'm less sure that the Cauchy integral formula is mentioned. For instance, I got no hits for "residue" in the book. So maybe the complex analysis part is restricted to a careful treatment of power series in $\mathbb C.$ – Dave L. Renfro Jun 21 '17 at 16:07
• Yes, it seems to me that that is how the final chapter is written. The progression of the chapters is fairly natural, except for the abrupt jump from metric spaces to analytic functions in the last chapter, without any discussion, it seems, of real differential calculus or of integration. The first 14 chapters of 15 cover roughly the same topics as Chapters 1-4 of Apostol, but in much greater detail and with more emphasis on formalizing the set-theoretic foundations of mathematics. – user49640 Jun 21 '17 at 16:33

The construction of the usual number systems is very explicit and clear in Classic Set Theory, which also happens to be an excellent first exposure to ZFC. Make sure you learn some category theory after playing with ZFC for awhile, to help give you new ways of looking at things that conflict somewhat with the ZFC worldview. Lawvere's book Sets for mathematics is good in this regard, and can be downloaded for free.

Here's an excerpt from my recommended book list. I think the biggest mistake a newbie at analysis can make is to be ambitious in their first book. Find the easiest rigorous book you can and master it. Then get a slightly harder one. Repeat.

"Yet Another Introduction to Analysis " by Victor Bryant is the book that I wish I had had when I was learning analysis, and if I was to write a book on the topic this is the way I would write it, (except that I won't because Bryant has already done it.) Bryant teaches analysis with lots of motivation and examples. The reader he has in mind knows calculus but cannot see the point of analysis. All mathematics is (or should be!) invented to solve problems and Bryant never forgets this, and explains why as well as how as he introduces each theorem. If you find analysis too dry, this is the book for you.

"Mathematical Analysis: A Straightforward Approach" by K.G. Binmore. If you find the jump from Bryant to Rudin too big, then Binmore is a nice in-between choice. This is actually the first book I read on analysis -- Bryant wasn't available at the time.

"Principles of Mathematical Analysis" by Walter Rudin. This a great second book on analysis. It starts from first principles but is drier that Bryant. So first read Bryant to get some idea of what is going on, and then work through Rudin to get all the details and to learn enough to prepare you for measure theory.

(the full list is on markjoshi.com)

• Thank you for the suggestion. Why exactly do you think that Rudin would be too hard? I do have some experience with proofs (also a bit analysis ones, not much though) – user370967 Jun 22 '17 at 10:41
• it's far too dry -- this is my considered opinion as student, teacher and researcher in analysis at some of the world's top universities. It will take less time to read Bryant, Binmore and Rudin than it will to read just Rudin. – Mark Joshi Jun 22 '17 at 10:44
• Thank you. These kinds of answers are the ones I was hoping for. – user370967 Jun 22 '17 at 10:45
• @Math_QED, kindly go through my answer as well. – user456218 Jun 26 '17 at 17:12

Ethan D. Bloch's The Real Numbers and Real Analysis is a fantastic book I used in college for my Real Analysis class, taught by professor Bloch himself. I highly recommend it, and if you need a list of some minor corrections made, please feel free to reach out.

I must recommend Ethan Bloch's The Real Numbers and Real Analysis. No more to say, just see the complete menu and the detailed preface from the author here! It's a quite good starting point into analysis.

ps: This book did astonishing well on real numbers and analysis of one variable that I haven't seen any well-known analysis book in amazon can reach its horizon; however, this book didn't cover the several variable analysis. If you want to study the latter, you need to find another book.

From the preface:

Multiple Entryways

A particularly distinctive feature of this text is that it offers three ways to enter into the study of the real numbers.
$$\ \$$ Entry 1, which yields the most complete treatment of the real numbers, begins with the Peano Postulates for the natural numbers, and then leads to the construction of the integers, the rational numbers and the real numbers, proving the main properties of each set of numbers along the way.
$$\ \$$ Entry 2, which is more efficient than Entry 1 but more detailed than Entry 3, skips over the axiomatic treatment of the natural numbers, and begins instead with an axiomatic treatment of the integers. It is first shown that inside the integers sits a copy of the natural numbers, and afeter that the rational numbers and the real numbers are constructed, and their main properties proved.
$$\ \$$ Entry 3, which is the most efficient approach to the real numbers, starts with an axiomatic treatment of the real numbers. It is shown that inside the real numbers sit the natural numbers, the integers and the rational numbers. This approach is the one taken in most contemporary introductions to real analysis, though we give a bit more details about the antural numbers, integers and rational numbers that is common.
$$\ \$$ The existence of three entryways into the real numbers allows for great flexibility in the use of this text. For a first real analysis course, whether for mathematics

Analysis by Its History by Ernst Hairer and Gerhard Wanner could be a good choice. The book is not only quite rigourous but also very entertaining.

I could recommend

1. Introduction to Real Analysis by Bartle and Sherbert
2. Mathematical Analysis by Binmore
3. Introduction to Classical Real Analysis by Stromberg

The first book is a very rigorous introduction to real analysis. The results are presented for $\mathbb{R}$. The style is somewhere between Spivak's Calculus and Bartle's out-of-print analysis.

The second book looks collection of lecture notes. The tone is conversational if you like those kinds of books. The book provides solutions of the exercises as well.

The third is my favorite. It does not assume any previous knowledge. Every real analysis book I have seen so far assume you are familiar with trigonometric functions, Euler number etc. Stromberg never uses these mathematica objects before defining. In my opinion, it is superior to the classical text of Rudin a.k.a Baby Rudin. To see what I mean compare the treatments of Cantor sets in both books. In fact, compare Stromberg with any real analysis book you will realize the difference.

I would recommend two books. The first is Introductory Real Analysis by Frank Dangello and Michael Seyfried . The second is An Introduction to Analysis by G. G. Bilodeau, P. R. Thie and G. E. Keough. These two books are alright for an introduction to single variable analysis.

If you want, try to look for an edition of Advanced Calculus by Watson Fulks or Advanced Calculus by R. Creighton Buck.

Finally, take a look at Introduction to Real Analysis by William F Trench. You can get it off the internet.

I would recommend that you pick up the book by Rudin along with a friend and try to read it together, slowly, replicating and demonstrating his proofs to each other. Rudin is a great book but it gets frustrating sometimes but don't try to bypass it. Also, it will also serve as a benchmark of your understanding of common techniques in analysis. When I first attempted Rudin, I used to find it very hard, but after a year when I looked back at it, I could solve most of it very easily. The other great advantage I felt because of doing Rudnin was that now whenever I read some advanced text in Functional or Harmonic Analysis and there is some lemma, a lot of times I can recall that he said something similar in a basic context using the very similar type of argument.

Tl:Dr; Find a comrade, do rudin, and don't give up.

• I have heard many people say now that Rudin is too hard to start with, so I'm considering to buy an easier book. I do not really have someone to read with as well :/ – user370967 Jun 26 '17 at 21:05
• In that case use Tao's book. He is a master. But then come back to Rudin. – user456218 Jun 26 '17 at 21:07
• (although I would say give yourself one week with Rudins 2nd chapter) – user456218 Jun 26 '17 at 21:07
• You suggest to try Rudin one week? – user370967 Jun 26 '17 at 21:10
• Yeah, I mean you have done calculus well so you should be able to do it given that you have sufficient will power. It is an intensely rewarding experience. If you feel that it is not going anywhere in 7 days you can pick up Tao's book. – user456218 Jun 26 '17 at 21:14

Be patient. If you continue as a mathematics major, you will take a course in your third year, "Advanced Calculus" that is a lot more rigorous. What you have taken is an introductory course intended to provide mathematical tools for physics, chemistry, engineering, and other technical professions. Be patient; it will become more interesting. A LOT more interesting, once the non-mathematicians are out of the class.

** Disclaimer: I am a registered professional engineer in California.

My old professor at UCLA, D.E. Weisbart, has a book "An Introduction To Real Analysis" which was pretty good---just to list something different.

Find it here.