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Ask HN: Resources to learn real analysis?
215 points by pedrodelfino 11 months ago | hide | past | web | favorite | 101 comments
Hi, I am an undergradute student of Applied Mathematics in Brazil. This semester, I will do a Real Analysis course and I am keen on learning this subject!

I love HackerNews. This is a great community with awesome people and marvelous content going on. It would be nice to receive some advice from you guys.

The professor is using the book "Analysis", from Terrence Tao. I am looking forward to supplementary material that will help me absorb this and gain some intuition.

1 - Is there a YouTube content particularly good for this topic?

2 - Is there some specific good strategy to study Analysis?

I really like to study doing exercises and, then, checking the answer. Not just the final answer but the whole answer.

This is not always available. Slader is a great website for that. Maybe there is an even a better resource than Slader that I do not know.

Thanks in advance!

Without criticizing any other people’s recommendations on this thread, I think it’s easy for people who have been doing proof-based math for a few years to recommend books that look clear and easy to them now, without remembering the time, effort and other support (e.g. great teacher and classmates) that may have been necessary to make use of that book a good experience.

Or maybe they are just way smart than me? :)

Either way, when considering possible books to use, I would ask the following question - is there a chance this is “too easy” to use / read, while still claiming to be about analysis? (I.e. calculus books fail this test because they don’t say they are about analysis). Then start with the easiest one unless there are really good reasons not to.

My own specific advice would be:

1) make sure you have had practice with proof based math before. If not, or you need the practice, get a copy of chartrand’s “introduction to mathematical proof” and do some exercises from the first 10 chapters. If you can do them easily, move onto analysis, if not, work through those 10 chapters first.

2) The book I personally like best for self-study is Abbott, “Understanding analysis” particularly if you can get the solutions manual, I think the explanations of the proofs are very good.

3) I would also recommend Lara Alcock’s book “How to think about analysis”, which is NOT a textbook, but has a lot of useful information and advice on how to learn analysis.

Also, obvious but worth repeating, if you are taking less than one hour per page to get through an analysis text, or don’t have pencil and paper in hand while going through the book, “ur doing it wrong” :)

One personal advice: if you want to learn a subject, don't focus on one book, one author or one set of suggestion. Only care about what makes your mind walk the domain.

I spent a decade with a terse text book on abstract algebra and went nowhere. It's for people who are either already enjoying concise math theorems or have the nack for it. I was missing a few bricks. A few years later some guy here or on reddit suggested a book that is vastly simpler, so simple it felt like HS but it cleared a few misunderstandings I had about notation and meaning. All of a sudden that 5$ ebay book had more value that my 100$ old paperbrick.

Which abstract algebra book if you don't mind me asking.

Sorry, first it's a linear algebra, it's by Gareth Williams, linear algebra with applications, I picked the 5th edition iirc, 5$ hardcover.

My 5c: both Fraleigh and Dummit & Foote are pretty good.

These come up all the time.

Yes, the math of real analysis is too difficult as a way to learn to read/write proofs. The easy way is a course in abstract algebra from a good teacher who is nice enough to read and correct early homework papers in proofs. Abstract algebra is usually just darned simple, e.g., proving things learned in grade school, so is a good place to learn about proofs.

As in my posts in this thread, before real analysis should be about three books in linear algebra, and that is also likely an okay place to learn to read and write proofs -- a course in abstract algebra with a good teacher as I mentioned is easier, still.

Sadly it's a fact that the academic computer science community has too many chaired, full professors who got their education in mostly just computer science, had few or no good math theorem proving courses, in their current work try to get deep into math with theorems and proofs, but, alas, consistently make serious mistakes in notation, how to state theorems, how to write proofs, etc. I saw the same thing, sad to see, from a EE prof working hard in coding theory. It shows. Apparently a person can get competent reading and writing proofs in some early, appropriate pure math courses or not at all.

Yes, real analysis, advanced calculus, differential equations, differential geometry, mathematical statistics, stochastic processes, etc. are way too difficult as places to learn to read/write proofs. Similarly even for more advanced material in linear algebra.

On the topic of newcomers to proof-based math - "How to Prove It" is a great resource: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...

This book made proof based math approachable for me. There are mechanical aspects to proofs that are probably best practiced in a structured way and this book does that by also acquainting you with set theory.

I have seen "How to Prove It" on Hacker News multiple times! I did a similar question about Discrete Math a month ago [1]. Someone also suggested this book. It is on the top of my "to read list". One of my friends says this book is his favorite one. [1] https://news.ycombinator.com/item?id=16355825

Real analysis was my first proofs-based math class and it really felt like being thrown into the deep end with no clue what was going on half the time.

I did quite like Bartle's "Introduction to Real Analysis" and "Elements of Real Analysis" books, kind of surprised that nobody else has brought them up. I think they strike an excellent balance between rigor and actually being comprehensible and approachable to people that aren't already familiar with proofs.

I think we make this harder than it has to be, by not doing proof-based math earlier.

Real analysis is often the first class in a math bachelor's program where things really get far from intuition and much harder to wrap your head around than the material from earlier classes.

Making that the first proof-based class is just piling one hard thing on top of another.

I think we'd be better off if we made basic calculus proof-based, at least for people such as math majors who are going to need to learn to follow and do proof-based math at some point. For those going into fields where they will not need to read and do proves, have a separate "practical calculus" track.

You can start out with more informal proofs at the start of basic calculus, and slowly step up the level of rigor throughout the year.

I can second the Chartrand book (I believe you mean "Mathematical Proofs: A Transition to Advanced Mathematics"). I used this book in a "bridge" class and loved it; it really starts from first principles and gives you a lot of simple examples that you can build on.

Yes, that is the correct title of the Chartrand book, thanks!

This is so true! I have found many textbooks very hard to work through. One you somewhat master the subject, the book seems exceedingly clear and you wonder why you ever struggled.

One thing that helps enormously in math is to have to hand in exercises to check your understanding. Making these exercises together (over a few beers, for example) helps to have some confirmation, reference, and fun.

I have learned real analysis by using the provided lecture notes. They were maybe 200 pages and I think this works better to get a grasp on the subject than using a textbook. If you use a textbook with 500 pages, you probably gonna skip or forget 75%. Lecture notes can be better tailored towards the background of the students and the contents of the course.

Also, I think an hour per page is a little long, but it just depends on the density of the textbook you're using.

Last thing, good luck, try to find a partner to work on problems, and don't despair: you might feel stupid at times but analysis is just hard.

+1 for "Understanding Analysis." Great book.

Tao's Analysis I is fantastic. There's a review available from the MAA: https://www.maa.org/publications/maa-reviews/analysis-i-0

The book doesn't touch on applications. Since you're studying applied math, you might want to supplement it with something like "Calculus with Applications" by Peter Lax and Maria Terrell.

On YouTube one can find lecture videos from a real analysis course given by Francis Su (former president of the MAA): https://www.youtube.com/playlist?list=PL0E754696F72137EC

Holy cow, check out that review: don't waste your time trying to teach this to an average undergraduate math major because it "would largely amount to swine facing pearls not meant for them."

It’s an allusion to a well-known bit in the New Testament, not actually calling undergrads swine.


I'm familiar with the expression. Its use here indicates that the reviewer believes the world is divided into those who are smart enough to follow the book and those who are wastes of time.

If it's not obvious why that sucks, imagine going to this professor's office hours if you're having trouble following a proof. Imagine that, rather than thinking you might just not have seen this style of proof before and trying to walk you through it, he indicates you aren't cut out to understand it and suggests you change your major.

(If it sounds like I'm being too harsh based on that one data point, after I read the review I went and looked up the professor on ratemyprofessor.com, and that seems to be his approach to struggling students: that he's weeding out people who just can't learn the material, not people who just got stuck because they're lacking the mathematical sophistication or exposure or confidence they need, or people who are learning more slowly and will be fine if they put in more time and effort and come up with better strategies. Not sure how a professor at Loyola Marymount decided he's qualified to be The Gatekeeper of Mathematics, but there it is.)

It doesn't really imply that. He just says, a bit stridently, that he thinks the book is targeted at people who will make maths the primary focus of their study. The other stuff is mostly you extrapolating a lot from a single offhanded remark. Reviews are not office hours. Going out to cherrypick additional 'data' doesn't really make your conclusion more sound. The dude wrote some sentence you don't like and next thing you know he's Vinz Clortho the Keymaster of Office Hours.

Well, if you're going to take the high road and give him the benefit of the doubt, I won't argue further. Maybe I'm just being cynical.

But in the review he wasn't just talking about "people who make maths the primary focus of their study." If he had just been talking about students' areas of interest or work ethic I would never have objected. It was specifically this: "One should pick one's audience carefully... and treat these gifted kids like apprentices." In my experience that approach misses a lot of talented people who were different enough not to get matched by the "gifted" filter.

most serious US math programs have honors sequences that are aimed at students who intend to go for a PhD and who are often starting with post-calc-I/II/III courses from the very beginning. tao says right in the introduction that this text is for such an honors sequence at UCLA. so, yeah, this book is explicitly written for undergrads who are being groomed to enter elite graduate programs because they did the work (before college) to fit that profile.

in the US, i believe the real analysis course for non-honors courses at R1s is often based on something like ross, abbott, or bartle & sherbert, whereas for non-R1s (where most math majors will be teachers) it may be based on something like lay or wade instead. these books are more accessible to students with less mathematical maturity.

i think what the reviewer is saying is that it would be a mistake to use this book in one of those non-honors courses with a poor faculty-student ratio. even if you do have some students who have the interest and ambition, you'd be doing a disservice to the rest. you'd be exceeding the level of interest for most and unable to support anybody adequately.

and people wonder why mathematics is reviled. talk about elitest.

I can also recommend Analysis 1. It's a pleasant book that's quite easy to read due to Tao's well-written prose.

I don't understand why so many people recommend baby Rudin (Principles of Mathematical Analysis). The presentation in Rudin is not merely terse, but also quite dry and unmotivated. I suggest you avoid it--regardless of how much talent or maturity you have. There are plenty of more interesting texts which will teach you just as much: Spivak and Pugh are nice, I also recommend the recent two-volume work by Zorich.

By the way, as you aquire experience you'll gain confidence and get over the urge to always check your answers. Here's a good exercise with a built-in answer key: When reading a text, every time you get to a result (claim, theorem, etc) try to prove it on your own before you continue. You probably should be doing that more often than not.

In any case, don't stick to just one text/source. Shop around, read a few pages here and there before you settle on something. There's no way a stranger on the internet can make a good recommendation: Find what works for you. The most important thing is that you're fully engrossed!

I've wondered that too. My conclusion is that it's mostly coming from folks who aren't distinguishing between something like elegance as a mathematical work and effective pedagogy.

The first person I met singing its praises was a hardcore linux guy who insisted on doing every task through a terminal with emacs—and this doesn't surprise me. I feel like there's a similar aesthetic at play here, and maybe a bit of fear that doing anything but the toughest option will make them weak (choosing these things on their own would be insufficient for that conclusion—but it often comes with a kind of scoffing attitude toward the 'lesser' options).

The logic behind toughest = most effective is a little confusing to me. Sure, grit has its uses in intellectual work, but getting effective instruction and building a solid foundation of concepts seems like it would outweigh it.

That's weird - I think Rudin is great and do everything through a terminal through emacs...

Haha—that's great!

But wait, you weren't in Berkeley, CA in 2013 were you?

I'm from Australia. My theory is that of the subset of people that are interested in both computer science and math, a significant portion use linux and if you use linux, then emacs is the best LaTeX editor (auctex and reftex are amazing). And "doing everything in emacs" is just what naturally happens when you use emacs long enough.

Makes sense. And so did your explanation for liking Rudin above (or wherever it's positioned now).

Question: I did EE degrees. Trying to fix possible gaps, I am going through Robert merlose notes on functional analysis. He refers to rudin for metric spaces. I am comfortable reading rudin, but I am hoping for an intuitive motivation for abstract definitions in general. Metric spaces have a physical motivation. What does one gain, if it is axiomatized? Maybe my bigger question is, in mathematics research, will abstraction always go towards symbolic manipulation and set theory. I am looking to avoid definition via axioms and motivate it. I hope I've explained myself.

You are asking big questions! My answer is brief but I hope it helps a little.

By axiomatizing a definition we can begin to prove theorems rigorously. If we formally abstract a concept and deduce formal statements from it (which can themselves be quite unituitive), we can be confident that the statements apply to the concrete situation at hand. On the other hand, if we always worked only with concrete or physical examples, we would have to figure out everything from scratch every time. A good general theory is one which concisely explains many specific cases at once.

For example: Can every periodic continuous function be approximated uniformly to whatever degree of precision we like by a partial Fourier sum? If we abstract away what's important, and study the situation in the general context of Hilbert spaces, we can answer this question not only for this specific case but also for a broad class of families of approximants in one fell swoop. We save a lot of effort by doing this, and we also usually gain extra insight into the problem.

Still, you don't want to abstract too early. It's important to understand the concrete case first before jumping a level in abstraction, otherwise you end up understanding nothing. If you're having any trouble with metric spaces (I'm not sure if you are) then you might find it helpful to look at a few specific examples to see what they're used for. Examples: In coding theory, we often use what's called the Hamming metric, the distance between two words. In graph theory, there is a natural geodesic metric: the shortest distance between two vertices as a walk along the edges. If that's not concrete enough, consider the popular "6 degrees of separation" rule: people are vertices and relationships are edges. Of course there are the usual examples of Euclidian space R^n, unitary space C^n, and the other various normed spaces you're studying in functional analysis.

By working with the axioms of metric spaces we can prove theorems which apply to all such cases at once. Here's an example of a tricky theorem: Let x be a point in a metric space. Suppose a subsequence satisfies the condition that each subsequence has a subsequence which converges to x. Then the entire sequence converges to x. You wouldn't want to prove this theorem from scratch in every specific concrete case! It's more efficient to abstract out the essential features (the axioms) and then prove the theorem in the general setting.

As for your other question: Set theory is (in my opinion) not a fundamental feature of mathematics. It just so happens that mathematicians today like to axiomatize everything in terms of set theory. Still, sets will always be useful even if we don't choose to define everything in terms of sets. A set is (loosely speaking) nothing more than a collection of objects, and we'll probably always find it useful to deal with collections of objects.

As for symbols: What more is a symbol than a hook on which we havg an abstraction? How can we manipulate abstractions if we don't have symbols for them? I'm using the word "symbol" in the broad sense here, which includes diagrams (or pieces of diagrams) and words in a natural language. If you want to see abstractions manipulated via diagrams and words, take a look at classical sources (say, pre-1500's). I believe our modern notation is far more amenable to manipulation and understanding.

In short: In mathematics, abstraction and symbol manipulation are the name of the game. If we have no symbols then we have no abstraction and no mathematics.

Rudin is a good reference book. It's not good to learn from, but if you're looking for the authoritatively _best_ proof, that cuts right to the heart of the problem, then Rudin is great.

It's not good for self study, but I used it as a supplement to a real analysis course I was taking at the time, and I appreciated the terseness and dryness. My lecturer went into the details, so it was good to have a terse book to remind me of lecture content when doing assignments and studying for exams.

I'm probably in a very slim minority of people who took their first topology course before they took their first real analysis course, and I think my understanding of both topics improved as a result of taking them in this order.

there's no way to actually avoid epsilon delta arguments in real analysis, but it's helpful to know that there is a more "intuitive" way of thinking about continuity (although admittedly it's a little weird when you first encounter it), that requires a lot less algebraic magic.

Anyway, there's more to real analysis than the topology of the real numbers, but I think it's a great starting place.

Hey! Not too long ago I did this myself. Real analysis is somewhat unique in mathematics for the intuitive writing it's attracted.

The most clearly written is John Baylis's 'What Is Mathematical Analysis?'. I _strongly_ recommend you read this. It's 125 pages of clarity and intuition building.

More rigorous alternatives would be: 1. Elias M. Stein's 'Real Analysis: Measure Theory, Integration, and Hilbert Spaces'. 2. Robert S. Strichartz 'The Way of Analysis'

Personally I would use the Baylis book and the other two as reference.

Additional resources include the top voted mathoverflow.com and math.stackexchange.com answers. Beyond useful.

Lastly, I have a twitter account (@math_twitr) that indexes (mostly) academic mathematicians. You might want to look through my follows there, they regularly post useful materials.

Addendum: Hm, in addition, I think you might want to look through this Amazon wishlist of mine consisting of those math books recommended for clarity by those who should know: https://www.amazon.com/hz/wishlist/ls/2B6H3IG4PS0R1?&sort=de...

Hi, I'm a maths student in Brazil myself :) I'll answer in english, though, since I'm not sure about the HN policy on comments in foreign languages.

Can I ask you which university you're from? I'm not aware of many universities besides UFRJ which offer an "Applied Maths" degree.

About real analysis, I took the summer course in IMPA, an I used only Elon's books. I really like them (for the books themselves, not just because they're in portuguese). There are two: the thick one: "Curso de Análise", and the thin, silver one: "Análise Real" (I like to call them Elão and Elinho :)). Elão is very detailed and has lots of examples, but mentions topics which may be too specific and not covered in your course. Elinho is much more terse, and great if you need a quick summary.

I would also consider reading David Bressod's "A Radical Approach to Real Analysis". It's an awesome book, which mentions historical motivations for everything, and has a really different approach to teaching analysis (it will certainly help you learn analysis, but might not help too much in your course, since it's quite non-traditional).

If you're not used to proof techniques, I highly recommend Keith Devlin's "Introduction to Mathematical Thinking".

About strategies to studying analysis: examples. I think it's really important to work out lots of examples by hand all the time. Every time you read a definition in your textbook, whatever it is, close the book and try to think of some examples of mathematical objects which satisfy the definition. When you're done, try to think of other examples which differ significantly from the ones you came up with before. When you open your book again, if the author presents examples, read them with attention. TLDR: as the other comments have made it very clear, you shouldn't be reading an analysis book without a pencil on your hand; you should feel active, not passive, while studying analysis.

Finally, I don't know about any youtube channels that could help you with an analysis course, but you should be aware of Mathematics.StackExchange. It's a great Q&A website/community; I've asked a lot of questions there while studying for my undergrad courses.

Wish you the best in your course and you maths career :)

Thanks, fofoni! I am actually doing a double degree in Law and Applied Mathematics at FGV, Rio de Janeiro. (I know, Law & Applied Math is weird rsrs). Maybe we should hang out someday. My email is p.delfino01 at gmail dot com, drop me an email!

Law+Math degree is definitely on the weirder side of things I've heard (and I endorsed Oxford's joint CS and Philosophy). But in that case I may allow myself to make an equally far stretch connecting something lawyerly with real analysis (that isn't about RA utility in data science for law enforcement). About how model theory shapes logic. It is advanced (practically algebraic geometry now) and not really related any more, but the basic issue came from set theory and analysis: models of infinitesimals, like in Keisler books. Since then it came to encompass and classify all logic-based mathematics (not every creative reasoning in mathematics is logical! though papers always are) and more exotic logics such as the „default logic” sometimes employed by lawyers.

It's Bressoud BTW, I endorse that too. Along with TW Körner „Companion to Analysis: A First Second and Second First Course” with Lang or Zorich as base. I wouldn't be as insistent on pencil at all times if it were to prevent broader reading or just expanded skimming.

To piggy back on this, Devlin has an excellent free course on Coursera roughly following the textbook.

Do NOT read Rudin. He is terse and unless you are already well versed in mathematics it is simply incomprehensible.

My recommendation would be Spivak’s calculus. There are a million great exercises and the book is beautifully typeset and overall a pleasure to read. Don’t let the title fool you, there are analysis exercises in there.

You have to decide between two things:

1 - you want to learn to prove things. Then yes, Rudin is a shitty text to learn by yourself because he really likes a certain type of, for lack of a better phrase, "beautiful" proof that requires a bunch of insightful jumps to get to. He'll then show the proof and really not discuss about how he got there. What a student needs is the ability to string facts/theorems that he or she knows together and how to turn that into a proof. Without a good professor, Rudin is (imo) terrible for that.

2 - you want to learn analysis, and care less about proving things. Reasons for this may be you need a bit the underpinnings for various reasons, and you care less about proving things and more about understanding. I think Rudin is a pretty good text then.

Spivak's calculus is a great book but be prepared to spend a lot of hours on it.

I LOVED Rudin. It was a long time ago, though.

At the very least, it has great exercises.

I also loved Spivak.

I highly recommend Francis Su's Real Analysis Youtube lectures, on Youtube [1]. He is an amazing teacher.

I first started trying to learn Real Analysis from Baby Rudin, but I couldn't understand the point behind the ideas introduced there. Then I started watching these lectures, which are based on Baby Rudin and mostly follow it, and it helped a lot (together with reading the main text itself - a crucial step).

The only bad thing is that only half of Rudin is covered - the other half is covered in Real Analysis 2, which is unfortunately not online as far as I can tell.

[1] https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F...

Since you are student at university, I would emphasize, in addition to your own reading: Talk to people who already understand the area. In office hours with your professor, or grad students, or even more senior undergrads. Just hang out with people who know more than you and talk to them about math, not even necessarily just analysis. It may not be the specific things they teach you, but rather the modes of thinking that are obvious after the fact but impenetrable before.

In addition to everything else you do, I recommend the following book:

"Counterexamples in Analysis" by Gelbaum and Olmsted.

You will find that many of your intuitions you picked up in calculus are violated in analysis. For instance, in calculus, many examples are both continuous and differentiable everywhere. But is every continuous function also differentiable? Nope! See the Weierstrass function [0].

The book is full of such counterexamples that will help you understand analysis at a deeper level (and avoid many pitfalls)

[0] https://en.wikipedia.org/wiki/Weierstrass_function

Really, for me most of my analysis knowledge is based on counter-examples. The counter-examples motivate the theorems and definitions, help check intuitions, and sometimes even help me remember the theorems and definitions.

When I took this course, several exam questions where of this form; give an example of a function that is something or satisfies something else. Reading this book would have helped.

I wasn't a math student, but I would probably look at the OpenCourseWare from MIT if I were trying to learn this stuff.

Analysis is 18.100 at MIT -- the variants are called 18.100A, 18.100B, 18.100C. There are further classes in the same vein, as well, such as 18.101.




I taught myself 18.100a from the book and website when I took it because class was too early in the morning and found them sufficient to make up for the missed lectures. Wonderful resources. Highly recommmend.

I'd highly recommend picking up 'Understanding Analysis' by Stephen Abbott for self study. Beginner friendly and easy to digest yet rigorous. I can't think of a specific strategy but trying to visually understand the core concepts like convergence, continuity etc could be of help.

I've watched a series of lectures on the subject by Harvey Mudd college, on youtube [1]. It helped me a lot, tho, I'd have to say these are introductory level, not really a deep dive, but more useful as an introduction.

[1] https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL04BA7A9EB...

May I suggest that you prepare mentally to read all the recommended books in the next ten years or so.

After you get into that mindset, pick one from the curated list and stick with it.

This will greatly enhance your experience with the book by reducing the anxiety that you might be reading the "wrong" book and missing out on the unicorn book out there, which you will read eventually.

In my experience, having that mindset (all vs mutually exclusive) diminishes the importance we give to the choice of the book, because we'll read them all. When the choice becomes less important, we spend time actually reading books instead of deliberating on which books to read.

My book recommendation include:

  - "A Course of Higher Mathematics" - V.I. Smirnov.
  - "Differential and Integral Calculus" - N. Piskunov.
  - "Problems in Mathematical Analysis" - Demidovich

Walter Rudin's book on Real Analysis is normally considered as classic in this area. However, you can also take a look at Mathematical Analysis by Tom M. Apostol, which helps in developing good intuition about the subject.

As others have mentioned, Tao has fantastic resources, and his blog might be worth looking at. Rudin is a classic. I studied real analysis through “Advanced Calculus” by Fitzpatrick which has some great material.

For more advanced analysis (esp. functional analysis) I would look at Kreyszig or Hunter & Nachtergaele.

The best way to prepare imo is to just do proofs between now and the start of the course. Try to find practice proof problems online and see if you can do them or find an entry-level book on discrete math. Problem-Solving Strategies by Engel is a good but slightly more advanced book for a beginner.

Abbotts Understanding Analysis.

I like the books from Elon Lages Lima ( in portuguese). The small one, 'Analise Real' has suggestions/answers for many of the exercises in the book.

Remember all the tricks and shortcuts you were taught when learning calculus?

Yeah, forget those.

Interesting. My courses on differential equations (PDE and ODE) were basically tricks. Very frustrating.

Ha! It took me 3+ years to really understand real analysis. How? I tried to imagine 10, 20, 30 etc examples for every abtract definition in the books. E.g. take the definition of open set. Try to imagine 10s of examples of open sets. Then try 10s of examples of closed sets. And similar. Then in your mind, you should develop the intuition that "open set is something that looks like one of these" vs "close set is something that looks like the others" etc. Then take the definition of continuous function, try to imagine every example possible! Just work on defn's with many, many, many examples. Henceforth, the theorems and proofs will become obvious... and presumably you'll end up being a good theorem-proof style mathematician.

No books necessary! If anything, I liked The Elements of Real Analysis by Bartle.

This is sometimes called inquiry-based teaching or Moore method, and… there is a textbook advertised as adhering to this particular style, Carol Schumacher's Closer and Closer: Introducing Real Analysis, haven't had it in hand though.

I have one thing to add. Whenever you meet a new example, check that example against your intuitions. When they mismatch, that is a great learning moment.

These also help in finding 'counter-examples' which tend to be very constructive.

This following a fantastic book, with good exercises, and most importantly it includes applications:

Real Analysis and Applications by Davidson and Donsig


Springer has problem books with full solutions:

https://www.amazon.com/Problem-Book-Analysis-Books-Mathemati... https://www.amazon.com/Problems-Solutions-Undergraduate-Anal...

The second one is for Lang's "Undergraduate Analysis" book.

Coursera's 'Introduction to Mathematical Thinking' is a great starter course for real Analysis https://www.coursera.org/learn/mathematical-thinking

I've spoken with more than one person who made it through Real Analysis intact by reading through "Introductory Real Analysis" by Kolmogorov and Fomin. There's a Dover version that you can probably find for $12 used... It was where I started, but I know several people who found it invaluable after struggling with other texts.

Introductory Real Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486612260/ref=cm_sw_r_cp_apa_a62T...

Come to think of it, there are a lot of good Dover books on mathematics.

Like another commenter said, "Understanding Analysis" by Abbot is a fantastic book. I am not mathematically inclined and this book got me an A- in my analysis course in undergrad. Rarely is mathematics so clearly explained.

I don’t have any resources for analysis directly to recommend that haven’t already been said, but there’s some good videos of Calculus by 3blue1brown on YouTube called The Essence of Calculus [1]. They are really well made and explain Calculus in a way that you get an intuitive feel for it. It may be helpful to learning analysis to understand Calculus really well, but I’ve never taken analysis so I can’t say for sure.

[1]: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...

Yes, 3Blue1Brown is amazing to get the idea behind and have revelation. At least in my case.

For math majors, baby Rudin is the standard. I would say though that it's probably too terse for most students. I like Pugh's book quite a lot. I think it strikes a good balance between not being long winded while providing enough explanation. Checking the whole answer as you mention is not going to be that helpful in general. There's more than one way to write most proofs and the nature of proof problems is that you have the answer at the beginning. Just make sure you're familiar with basic proof techniques.

Others here have offered some great suggestions already, so I will offer one a little off the beaten path:

Foundations of Mathematics https://www.amazon.com/Foundations-Mathematics-Ian-Stewart/d...

This book is meant to help one transition from performing math in an algorithmic manner to generating proofs based on logic and also set theory.

Don't know if anyone has suggested it here (I'd honestly be surprised if someone hasn't) but Principles of Mathematical Analysis by Walter Rudin, affectionatey known as "Baby Rudin" is a classic book. It's known for being relatively difficult and dense if you're just beginning with analysis, but if you go through the book and complete a fair number of exercises, it's an incredibly rewarding experience and definitely grants a ton of mathematical maturity.

Digression: Terry Tao wrote a book on analysis? That's awesome!

Recommendation: https://www.amazon.com/Introduction-Analysis-Dover-Books-Mat...

There's a Dover book called "Introductory Real Analysis" by Kolmogorov & Fomin. It's one of Richard Silverman's translations from the Russian. It's got a few typos in it and the feel is a little old-timey, but the mathematical content is beautifully laid out. Read it for culture and a look at the bigger picture. It should be a good complement to Tao's book.

Also Shilov.

Good question ... I haven't seen a great course on real analysis on Youtube/MOOC platforms. If anyone has recs, I would also be interested.

For sure, there’s only so much a lecture in RA can achieve. Ultimately the student has to write rigorous proofs, and lots of them.

Rudin's classic texts are a great resource.

Rudin is a good breviary rehearsal if one already almost-knows and feels the material. People with certain inclinations may get a warm fuzzy feeling how things neatly fit together as if by omniscient design. Otherwise it is a crossword puzzle to amuse the god himself.

You may get a feeling you understood things (and earned that), but you are wrong. When I ask people what they really remember Rudin from, what specific piece of proof they learned specifically from his main and baby books, I hear only vague answers. There is a pretense of “teaching to think” by omission, also common in some dated textbooks, but I personally would leave that to professionals over at philosophy dept. and focus on clear exposition leaving neatness for examples. OTOH if you are predisposed to lauding yourself for how smart you (or Rudin) are for figuring all the tricks (knowledge of which stays at this level), you are in for ego boost (or bust).

Especially for real analysis his treatment of Lebesgue integral is worse than just about anyone else's I know (also, in 21st century it's time for better integrals like e.g. Henstock–Kurzweil). The only thing worse is again Rudin's own treatment of differential forms.

In professional mathematics Rudin is renowed for numerous many things, among them the Rudin–Keisler order in the theory of ultrafilters and ultraproducts. It is a sign about reading order. Because as it happens Keisler also wrote a textbook, and from a diametrically opposed perspective. Equally far fetched in the other direction, one of intuitionistic non-standard infinitesimals. I think being a product of certain totalising era of uniformisation in mathematics these texts are complimentary.

For a reader interested in somewhat extended real analysis I would recommend Lang, Bressoud, Körner („Companion…”).

Mentioned Terence Tao book from weblog-notes for his original RA course is also freely downloadable as pdf.

Finally Strichartz is overly chatty wordier antithesis to Rudin.

The introductory one is known as Baby Rudin and is a pretty good first exposure to real analysis.

This one? "Principles of Mathematical Analysis"

Yup. You can get the international version for cheap: https://www.amazon.com/Principles-Mathematical-Analysis-Rudi...

The fact that US version of this book from 1976 remains over $100, while the international version is under $10, and the Kindle version is over 50% more than the paperback, is such a perfect example of everything wrong with the textbook market.

This is how I learned it! (Baby Rudin, anyway.)

I wonder if this is still the go-to for undergraduate classes --- does anyone know?

Yes, last year at my university we used Baby Rudin for the undergraduate real analysis sequence, and Tao's books for the graduate level sequence.

+1 I am currently reading this. It’s dense but very clean.

I'm surprised no one mentioned this yet: make sure to talk to your professor, frequently. Introductory analysis courses exist primarily to teach a certain way of thinking, and there is, after all, no better way to learn how to think than to talk to someone who already knows the ways. Take advantage of what you have.

You might check out Polya and Szebot: Problems and Theorems in Analysis: two volumes. Old (original German edition 1925), but by one of the great mathematicians of the twentieth century and a longtime collaborator. You might also like Polya's "How to Solve It", a true classic.

An interesting supplementary book is "Analysis by Its History" that gives insight into how classical analysis was actually developed. Not so great to learn from initially, but gives some background on the intuitions from which the modern definitions are based.

your strategy of working exercises is a good one. it can also apply to results in the text: attempt to prove the result independently before reading the proof supplied in the text. if you find this practice enjoyable, it might be a sign that you'd like to study pure math.

texts: carothers for reading like a novel, rudin for taking apart like a car engine...

... or, drop all your classes and learn to formulate everything in the terms of measure theory from the beginning. halmos's texts on any (mathematical) subject are almost always well balanced ...

... o! that reminds me. also, there's this thing called "functional analysis" that'll be worth looking into after basic topology is well-cemented.

I think a lot of responses here are forgetting that OP is an undergraduate student in an _Applied_ Mathematics course. "A Concise Introduction to Pure Mathematics" by Martin Liebeck would be my suggestion.

One thing that helped me was to draw pictures both before and during proofs.

I would recommend baby Rudin, but it was only after a course in real analysis, measure theory, and functional analysis that I was able to come back to it, understand it and appreciate it.

Since you are in Brazil, Elon Lages Lima "Curso de Análise".

Draw lots of pictures. That habit helped me learn analysis most.

I hear that libgen.io is a useful resource for books...

Part I

One way and another, I got a good background in real analysis. So, okay, I'll try to answer:

An answer depends on what is meant by real analysis.

Part of the answer is advanced calculus, and part of that is the Gauss, Green, and Stokes theorems. If do these the modern and high end ways, then get as deep as like, and spend as much time as like, in differential geometry, calculus on manifolds, differential forms, E. Cartan, exterior algebra, algebraic topology, etc. But for 19th century physics and engineering, there is a way to get a good treatment of what need in about a nice weekend from (own TeX markup):

Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced Calculus,\/} Addison-Wesley, Reading, Massachusetts, 1957.\ \

Note how old it is! It's no longer really "modern"! Get a used copy; that's what I did. You don't want a newer edition!

But with that out of the way, might take a fast pass through an old MIT standard

Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \

There's a lot of fun stuff in there, but it's TOO short on proofs. So, I'd pass it up and return later when know enough real analysis to guess or understand most of the proofs easily.

Note: In (the relatively elegant and easy to read)

George F.\ Simmons, {\it Introduction to Topology and Modern Analysis,\/} McGraw Hill, New York, 1963.\ \

the flat statement IIRC "The two pillars of analysis are linearity and continuity."

Well, for the linearity, really need a good background in linear algebra. For this, you need at least three books, a really easy one, elementary, that starts with, say, systems of linear equations and Gauss elimination. Then a more advanced one that emphasizes the axioms for a vector space, does vector spaces with at least both of the real and complex numbers (over finite fields can be important for computing but not for real analysis) and emphasizes eigenvalues and eigenvectors, and finally the grand one, written at the knee of von Neumann as a finite dimensional introduction to Hilbert space theory,

Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition,\/} D.\ Van Nostrand Company, Inc., Princeton, New Jersey, 1958.\ \

If you get very far in real analysis, then you will want a good treatment of at least basic Hilbert space theory, and Halmos is a good start.

For his section on multi-linear algebra, I'd skip that unless plan to take the exterior algebra of differential forms seriously. In that Halmos book, concentrate on vector spaces, vector subspaces, linear transformations, eigenvalues, eigenvectors, and Hermitian and unitary matrices. Also at the end note the cute ergodic theorem! The big deal in about the last half of the book is spectral decomposition -- don't skip that.

For your second book, I used E. Nearing -- he was a student of E. Artin at Princeton. So, Nearing's book is high quality stuff. I worked carefully through that and learned a lot. But his appendix on linear programming is a disaster! Can do nearly everything important in linear programming and its simplex algorithm as just a simple -- learn it in an hour -- extension of Gauss elimination. Nearing does finite cones and dual cones for which he never makes a clear connection with linear programming. And although he works with all those cones, still he misses the theorems of the alternative -- Farkas, etc. -- important in parts of optimization, convexity, etc.

Also recommended is Hoffman and Kunze and, IIRC, available for free on the Internet as a PDF file.

There is much more in linear algebra, e.g., from R. Bellman, R. Horn, on numerical methods, etc. but these are not crucial for a rush to real analysis.

Maybe part of real analysis, that is, advanced enough, is "Baby Rudin", Principles of Mathematical Analysis. The later editions have near the end some tacked on material, mostly without sufficient context, on the exterior algebra of differential forms. Skip that. If you want that material, then go for R. Buck, Advanced Calculus or Spivak, Calculus on Manifolds or really just go for a real book on differential geometry, manifolds, calculus on manifolds, etc. Such differential geometry is from important to crucial for several objectives but is NOT on the mainline of a rush to real analysis.

So, what is going on in Baby Rudin? Okay, the main idea of the book is that we can give a solid development of the Riemann integral for a function that is continuous on a compact set. So, get to learn about continuity, that is, one of the two pillars of analysis. Then hand in hand with continuity is compactness, so get to see that. All of that is in just the first few chapters; that's what those first few chapters are all about -- continuity and compactness. E.g., get to learn that in R^n (for the set of real numbers R and a positive integer n), a set is compact if an only it is closed and bounded -- super, important, crucial stuff, the key to a clean up of Riemann integration, that is, material Newton didn't know.

Then with that material on continuity and compactness, Rudin does the Riemann–Stieltjes integral. There, mostly just ignore the Stieltjes part with its possibility of step functions (maybe as a cheap answer to what the physics people try to do with the Dirac delta function, which, of course, is not really a function, but has a clean fix-up with distributions and measure theory) and just read that Rudin material for the Riemann integral of first calculus. You will likely never see the Stieltjes extension again.

The main idea is: A function continuous on a compact set is also, presto, bingo, wonder of wonders, really nice day, uniformly continuous, and that makes the derivation of the Riemann integral really easy. Really, that's the core idea of the whole book. Baby Rudin can seem severe, but with this introduction you should be able actually to like it a lot. Later in the book, Rudin touches on the fact that the uniform limit of a sequence of continuous functions is continuous -- same song, next verse. It was a Ph.D. qualifying exam question for me; I did get it.

Later he gives a really nice treatment of Fourier series -- that is very much worth reading.

I'd suggest one side trip: Cover the inverse and implicit function theorems. They are just a local, non-linear generalization of what you will see really easily for linear transformations in linear algebra via, right, just Gauss elimination. For a source? There is a good treatment in W. Fleming, Functions of Several Variables. IIRC, there is also a cute proof based on contractive mappings.

So, by then you will have a good start on both continuity and linearity.

Part II

Starting there, as you go on in real analysis, you will place much less emphasis on continuity. The main work will replacing the Riemann integral by something that does much better on the edge or pathological cases.

Well, the Riemann integral was, right, a lot about area. Well, it's actually not a really good theory of area -- doesn't have all the nice properties we would wish for. So, that view of area gets polished up and replaced by measure theory. Next, that theory of area is used to define the Lebesgue integral, which is much nicer than the Riemann integral, e.g., gets rid of the assumptions of a compact set and continuity.

Lebesgue was a student of E. Borel and did his work near 1900. You will see the Heine-Borel theorem in Baby Rudin.

In simplest terms, the Lebesgue idea is to partition on the Y axis instead of the X axis like the Riemann integral does. Just why that works so well is cute. It also frees up the domain of the function to be much more general, and in 1933 A. Kolmogorov used that fact, finally, to give a good foundation for probability theory, the one accepted now in essentially all advanced work in probability, stochastic processes, and mathematical statistics.

Likely the nicest book to read on real analysis is H. Royden, Real Analysis. But also, later, read the first, real half of W. Rudin, Real and Complex Analysis.

Note: For real analysis, complex valued functions of a real variable are important; functions of a complex variable are quite different and not important; Rudin has some deep reasons to disagree with me. Rudin is no doubt correct; for students trying to learn, I'm correct!

Once you have the main ideas in mind, and maybe what I've given here will be enough, Rudin is more succinct and nicer to read than Royden. But read them both, Royden first. Go through Royden quickly since will do it all again in Rudin.

In Royden, notice his Littlewood's Three Principles as a cute view of what is going on in measure theory and the Lebesgue integral. Royden also has some exercises on upper and lower semi-continuity -- glance at those and get the main idea for the remaining connection with continuity. Notice the nice treatment of differentiation -- the connection between integration and differentiation is weaker for the Lebesgue integral because the integral is so much more general. Then notice the Carathéodory extension result -- that is the key to defining measures on the real line, in particular, Lebesgue measure.

Spend at least a weekend in

John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \

An amazing weekend.

Keep at hand

Bernard R.\ Gelbaum and John M.\ H.\ Olmsted, {\it Counterexamples in Analysis,\/} Holden-Day, San Francisco, 1964.\ \

Notice that for the Lebesgue integral, there is one, central proof technique in just four steps: Prove the theorem for a function that is just a positive constant defined on a set, say, just a box. By linearity, generalize to finite sums of such functions. Then by monotone continuity (see early on the monotone convergence theorem), prove the result for all non-negative functions. Last by linearity again, get the result for all integrable functions.

Okay, for integrable, there is a cute approach, trick: Define the integral for non-negative functions, even ones defined on the whole real line or all of R^n (for the set of real number R and a positive integer n) or an abstract measure space. Okay, easily enough, that integral can have value positive infinity (the Riemann integral gets sick here). Then do the same for a function that is all <= 0. Then given a function, write it as the sum of its positive part and its negative part. Integrate those two parts separately. Now, if at least one of those two integrals is finite, then add them for the integral of the given function. So, we are okay in all cases except where the positive part integrates to positive infinity and the negative part integrates to negative infinity -- we don't want to subtract infinities because permitting that ruins the laws we want for arithmetic.

For this four step proof technique, can knock off a really nice version of Fubini's theorem, that is, interchange of order of integration.

The Lebesgue integral with the dominated convergence theorem, which will see early on, give nice versions of differentiation under the integral sign.

Note: Commonly in physics, engineering, probability theory, etc. we write integrals over the whole real line. Alas, really, for Riemann integration, the appropriate theorems are rarely presented; they are not in Baby Rudin. Really, the Lebesgue integral is needed. So, you've been needing the Lebesgue integral for a long time.

Then get to apply the Lebesgue integral to the powerful Radon-Nikodym theorem (Rudin gives von Neumann's cute proof based, amazingly, in part on just polynomials), get the main, important duality theorems (the Lebesgue integral is the main linear operator!), Banach spaces, Hilbert spaces, and the Fourier integral.

Cover that and can claim you have a good start on real analysis. You will also have apparently so far the only good background for probability, stochastic processes, and mathematical statistics. E.g., the Radon-Nikodym theorem is crucial for conditional expectation, Markov processes, martingales (amazing things with one of the strongest inequalities in math and a super short proof of the strong law of large numbers), ergodic theory, sufficiency in statistics, a quite general proof of the Neyman-Pearson result in statistical hypothesis testing, and more, e.g., the role of Brownian motion in potential theory, stochastic optimal control, ....

Go for it!

Royden is a classic - my mom and I both used it in our grad math programs, almost 30 years apart or so.

Thanks for your in-depth posts!

I worked through 9 chapters of Apostol.

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