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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!
- corporateguy5 9y agoPrinciples of mathematical analysis by Rudin
- roscoebeezie 9y agoOne thing that helped me was to draw pictures both before and during proofs.
- VideoEveryDay 9y agoRudin's classic texts are a great resource.
- harpocrates 9y agoThe introductory one is known as Baby Rudin and is a pretty good first exposure to real analysis.
- throwawaybbqed 9y agoThis one? "Principles of Mathematical Analysis"
- mugamma 9y agoYup. You can get the international version for cheap: https://www.amazon.com/Principles-Mathematical-Analysis-Rudin/dp/1259064786/ https://www.amazon.com/Principles-Mathematical-Analysis-Rudi...
- opencl 9y agoThe 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.
- greenleafjacob 9y ago+1 I am currently reading this. It’s dense but very clean.
- thraway180306 9y agoRudin 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.
- deleted 9y ago[deleted]
- asafira 9y agoThis is how I learned it! (Baby Rudin, anyway.) I wonder if this is still the go-to for undergraduate classes --- does anyone know?
- snakeboy 9y agoYes, last year at my university we used Baby Rudin for the undergraduate real analysis sequence, and Tao's books for the graduate level sequence.
- deleted 9y ago[deleted]
- throwawaybbqed 9y agoGood question ... I haven't seen a great course on real analysis on Youtube/MOOC platforms. If anyone has recs, I would also be interested.
- ianai 9y agoFor sure, there’s only so much a lecture in RA can achieve. Ultimately the student has to write rigorous proofs, and lots of them.
- trentmb 9y agoAbbotts Understanding Analysis.
- jhck 9y agoTao's Analysis I is fantastic. There's a review available from the MAA: https://www.maa.org/publications/maa-reviews/analysis-i-0 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 https://www.youtube.com/playlist?list=PL0E754696F72137EC
- aspaceman 9y agoI can also recommend Analysis 1. It's a pleasant book that's quite easy to read due to Tao's well-written prose.
- tashi 9y agoHoly 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."
- mlevental 9y agoand people wonder why mathematics is reviled. talk about elitest.
- pvg 9y agoIt’s an allusion to a well-known bit in the New Testament, not actually calling undergrads swine. https://en.wikipedia.org/wiki/Matthew_7:6 https://en.wikipedia.org/wiki/Matthew_7:6
- tashi 9y agoI'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.)
- stiff 9y agoSpringer has problem books with full solutions: https://www.amazon.com/Problem-Book-Analysis-Books-Mathematics/dp/1441912959/ref=sr_1_2?ie=UTF8&qid=1521910574&sr=8-2&keywords=analysis+problem+book https://www.amazon.com/Problem-Book-Analysis-Books-Mathemati... https://www.amazon.com/Problems-Solutions-Undergraduate-Analysis-Mathematics/dp/0387982353/ref=pd_sim_14_2?_encoding=UTF8&pd_rd_i=0387982353&pd_rd_r=0W0RXBR8HFR8EV0WNY5J&pd_rd_w=BdfFH&pd_rd_wg=fUEbl&psc=1&refRID=0W0RXBR8HFR8EV0WNY5J https://www.amazon.com/Problems-Solutions-Undergraduate-Anal... The second one is for Lang's "Undergraduate Analysis" book.
- gbacon 9y agoRemember all the tricks and shortcuts you were taught when learning calculus? Yeah, forget those.
- pedrodelfino 9y agoInteresting. My courses on differential equations (PDE and ODE) were basically tricks. Very frustrating.
- meiralins 9y agoI 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.
- superquest 9y agoDraw lots of pictures. That habit helped me learn analysis most.
- johnsonjo 9y agoI 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-rj53DwVRMYO3t5Yr https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...
- praeconium 9y agoYes, 3Blue1Brown is amazing to get the idea behind and have revelation. At least in my case.
- matheus2740 9y agoI'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=PL04BA7A9EB907EDAF https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL04BA7A9EB...
- ghufran_syed 9y agoI hear that libgen.io is a useful resource for books...
- ianai 9y agoI worked through 9 chapters of Apostol.
- kanishkalinux 9y agoWalter 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.
- wish5031 9y agoAs 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.
- ouid 9y agoI'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.
- ghufran_syed 9y agoWithout 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” :)
- defen 9y agoOn 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/0521675995 https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...
- jfv 9y agoThis 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.
- pedrodelfino 9y ago
- sonabinu 9y agoCoursera's 'Introduction to Mathematical Thinking' is a great starter course for real Analysis https://www.coursera.org/learn/mathematical-thinking https://www.coursera.org/learn/mathematical-thinking
- colejhudson 9y agoHey! 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=default https://www.amazon.com/hz/wishlist/ls/2B6H3IG4PS0R1?&sort=de...
- auntienomen 9y agoThere'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.
- thraway180306 9y agoAlso Shilov.
- throwaway080383 9y agoSince 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.
- trendia 9y agoIn 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 https://en.wikipedia.org/wiki/Weierstrass_function
- marktangotango 9y agoWhen 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.
- rocqua 9y agoReally, 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.
- makintunde 9y agoI 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.
- yantrams 9y agoI'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.
- jfv 9y agoDo 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.
- x0x0 9y agoYou 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.
- AlexCoventry 9y agoI LOVED Rudin. It was a long time ago, though. At the very least, it has great exercises. I also loved Spivak.
- graycat 9y agoPart 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.
- ninguem2 9y agoSince you are in Brazil, Elon Lages Lima "Curso de Análise".
- totalZero 9y agoI 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. https://ocw.mit.edu/courses/mathematics/18-100a-introduction-to-analysis-fall-2012/ https://ocw.mit.edu/courses/mathematics/18-100a-introduction... https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010/ https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-f... https://ocw.mit.edu/courses/mathematics/18-100c-real-analysis-fall-2012/ https://ocw.mit.edu/courses/mathematics/18-100c-real-analysi...
- steindavidb 9y agoI 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.
- spacedome 9y agoAn 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.
- edanm 9y agoI 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=PL0E754696F72137EC https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F...
- phaet0n 9y agoThis following a fantastic book, with good exercises, and most importantly it includes applications: Real Analysis and Applications by Davidson and Donsig http://www.math.uwaterloo.ca/~krdavids/RAA/real.html http://www.math.uwaterloo.ca/~krdavids/RAA/real.html
- fofoni 9y agoHi, 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 :)
- pedrodelfino 9y agoThanks, 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!
- joebergeron 9y agoDon'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.
- corndoge 9y agoLike 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.
- haZard_OS 9y agoOthers 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/dp/019870643X/ref=br_lf_m_fa2gmqeu35du8k8_img?_encoding=UTF8&s=books 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.
- jinfiesto 9y agoFor 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.
- yshklarov 9y agoI 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!
- westoncb 9y agoI'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.
- monaghanboy 9y agoDigression: Terry Tao wrote a book on analysis? That's awesome! Recommendation: https://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383/ref=sr_1_2?s=books&ie=UTF8&qid=1521948380&sr=1-2&keywords=real+analysis+dover&dpID=41p-0AKJ7lL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch https://www.amazon.com/Introduction-Analysis-Dover-Books-Mat...
- starchild_3001 9y agoHa! 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.
- rocqua 9y agoI 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.
- thraway180306 9y agoThis 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.
- auvrw 9y agoyour 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.
- MaysonL 9y agoYou 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.
- Jugurtha 9y agoMay 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
- chaboud 9y agoI'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_a62TAbNFPVY5M 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.
- markhkim 9y agoI'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.
- uptownfunk 9y agoI 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.