Ask HN: Resources to learn real analysis?
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!
104 comments
[ 25.7 ms ] story [ 216 ms ] threadYou 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.
I wonder if this is still the go-to for undergraduate classes --- does anyone know?
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
https://en.wikipedia.org/wiki/Matthew_7:6
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.)
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.
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.
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.
Yeah, forget those.
[1]: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...
[1] https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL04BA7A9EB...
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.
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.
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” :)
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.
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.
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.
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.
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.
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...
"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