The Real Analysis Survival Guide
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- Опубликовано: 27 июл 2024
- How do you study for Real Analysis? Can you pass real analysis? In this video I tell you exactly how I made it through my analysis classes in undergrad and graduate school, and then moved on to become a professor of mathematics.
//Books
Rosenlicht (short and easy to follow) - amzn.to/3k9X2Zc
Rudin (gold standard, but a bit opaque) - amzn.to/3KeL0IC
Apostol (thorough, but long) - amzn.to/3LhEylx
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0:00 Introduction
1:20 The Best Books for Real Analysis
3:55 Chunking Real Analysis
5:30 Sketching Proofs
6:40 The key to success in Real Analysis
Most of my exposure to analysis is through self-study, and one of my favorite books by far is "Introductory Real Analysis" by Kolmogorov and Fomin. Very challenging text, but also immensely satisfying to go through! Honorable mention goes to Shilov's "Elementary Real and Complex Analysis," which, although has a handful of typos throughout (most of which will be obvious to the cautious reader), breaks down a lot of the more complicated topics into simple, easily digestable explanations and examples.
Both are also Dover reprints, so they're dirt cheap to pick up.
I had a similar experience with real analysis when I took it. I had first taken abstract algebra and finished the course with a 99.4% average. Real analysis seemed to start out easy, but as the course progressed, I realized the teacher had very specific ways to prove things. I wasn't allowed to think outside of the box and come up with my own method of writing proofs. I really had to reflect on what was being accomplished with epsilon delta proofs. After getting a deeper understanding, my grades started improving and just managed to finish the class with 93.7% which is an A at my institution.
Everything is spot on. As far as texbooks go, I really like Abbot and Tao’s Analysis 1 & 2. I have one caveat though. I agree that one should avoid looking at the solution until you’ve given the problem your best shot. However, for the self-learner who doesn’t have a professor to provide hints and/or give feedback to your work, solutions manuals become really useful. Otherwise, how do you check your work? Feedback is essential to improve your craft in mathematics, otherwise you’ll just repeat mistake over mistake and that’s not efficient learning
I am currently learning Analysis out of Ross’s ‘Elementary Analysis: The Theory of Calculus’ and its very intermediate in terms of difficulty so far. Not outright easy, but definitely a lot easier than I thought. Maybe because I have gotten a lot more mathematical experience this year and spent lots of time doing number theory proofs. Because I have noticed that Analysis proofs have a similar flavor to number theory proofs.
Usually a lot of number theory proofs tend to revolve around divisibility relationships and well-ordering. Also a lot of mathematical induction. And those skills I learned seem to have carried over well into many epsilon proofs of all types which you would do in Analysis. Just my own experience so I don’t know if that holds in a more general case for everyone. But I do feel that having that type of preparation in specific types of proof writing skills which involve lots of algebraic manipulations and set construction based on the well-ordering principle can help a lot for Analysis.
I've taken the first and second undergraduate course on real analysis already but this video was still extremely helpful, especially the idea of chunking. I will definitely apply these techniques when studying real analysis again prior to enrolling into graduate school!
Thank you! Suggestions are very helpful.
Your dedication to repetition really dawned on me! That's the key to success. Thank you, Professor!👍
It really is. When it clicked for me, everything changed. Best of luck with your studies!
I think when talking about Rudin we should not forget his celebrated book Fourier Analysis on Groups (actually locally compact abelian group). When writing the trilogy he was certainly keeping this book in mind. FAG should be accessible after Part 3 of Rudin's Functional Analysis (for the record, part 2 and part 3 are independent of each other; one can jump to part 3 right after finishing part 1).
A great thing about his PMA is how he wrote the book from a modern mathematics POV. Banach space, differential form, and topology on metric space, all of them are 20th century things.
Rudin also invented a word during his era: lgbalcag (Let G be a locally compact abelian group).
loved this video, keep making videos like that! ❤️❤️❤️
Thanks! I’ll do my best :)
It's really strange when people say to avoid doodling in Analysis when a lot of concepts are clearly "visual" metaphors. An open BALL, a CONNECTED set, a finite COVER and so on. Of course, the idea is that these concepts are more abstract than what we can doodle, however I'd argue that doodling isn't a handicap to the "abstraction step" as long as the student knows that these visualizations are just instruments in order to understand things better. Easily you could explain why the open ball isn't exactly a ball... just show what happens when we use the Manhattan distance/l1 norm instead of the euclidean distance.
I completely agree. When I was an undergrad, I just listened to my professors and tried to do everything without visualizing. I’m contrast, a friend of mine doodled everything out. I realized before long they he had much better intuition than I did, and so I started changing my habits.
Great advice on chunking. Often there's little time to do this important step - having to work to make ends meet and other classes detract from the ideal. Little wonder why math majors are sleep deprived. Prior to taking real analysis I audited the first week and midway through the semester asked a friend taking the class how things were going. The class started out with ~25 students and midway through it had 5 students left. Some of them thought they had missed some prerequisite/required class. In times past, there was absolutely no "how to prove" class. It was just the sequence of Calc, one semester of Lin. Alg and ODE each, then analysis with Baby Rudin. It was sink or swim with proofs. It was not surprising that a lot of potential math majors switched to CompSci after this class.
Yay, a new (for me) math channel with interesting content.
Happy to have you here!
Great video, and extremely helpful!
Thank you! I’m glad you found it helpful!
Absolutely agree with you, cartoons/sketches helped me better understand and remember proof strategies, and also made me mindful of their shortcomings, ofcourse. Sketching specifically helps for topics such as compactness, continuity, Cauchy sequences, and 'dense subsets', where it's very helpful to visualise things (atleast at the beginning stages of the learning process).
I have a weird quirk I think I picked up from my PhD adviser. If a set is open, then I draw it as a nice blob, but if it is compact, then it's a polygon with sharp edges.
I convert almost every proof to a visualization in my head. Its not a substitute for formal statements, but it helps with working memory. I don’t know how people find the chained inequalities you need for delta-epsilon proofs without visualizing number lines. I can use pure algebraic manipulation to check results, but it doesn’t help me find the trick I need to put the correct bound on a variable to show some kind if convergence.
Great video !
I think this video can be helpful to anyone who wants to learn proof based math.
This video was so informative! Would love to see The Functional Analysis Survival guide!
Someday! Functional Analysis can be of so many different flavors
Very nice advice. Thanks!
My pleasure!
Your videos are really inspiring.
That means a lot to me, thank you 😊
Thanks and awesome advise
My pleasure!
Boomer here, Math degree 85, re-vamping now with a focus.
There are infinite refinements, yet you can focus, such as on ability to compute and how to do that with good explanatory resources such as Jay Cummings's - no association here in any manner but like the approach for student's. Note my focus from previous posts.
I can also recommed "A primer of real functions" by Boas, it is a really nice introductory book
I’ll have to check it out. I’ve only read Boas’ Entire Functions.
The best undergraduate textbook is Stromberg, “introduction to classical real analysis “
What makes this book the best in your opinion?
@@soyoltoi author name is super cool ofc
My favourite is introduction to Real Analysis by Bartle and Sherbert
In India we have study in our 1st semester. After working on basic calculus we come to an abstract pure mathematics. That is one of the hardest and I would say it is so fun to study if you understand the basics
Real analysis from J.Cummings is really the best real analysis book to me, followed by analysis from T.Tao.
It's rigorous and cover all the necessary topics, yet it really feels like you read a human-made text (not some kind of low-level source code that you should compile :D), all the proofs are written in such a way that you can understand WHY we prove it that way, and what is the meaning of the concepts at play ...
Really i cannot recommend this book enough, it's a pleasure to read.
T.Tao is in the same spirit and it covers a LOT more content, but its a bit more formal to me.
Apostol is great. It covers more general topics than other books..from real line to abstract spaces and higher dimensional Euclidean spaces. It is also hard though for beginners
Yeah, I agree. Apostol is still a tough book for beginners. But on the scale of Rudin, Apostol, and Rosenlicht, I feel it fits squarely in the middle. Thorough coverage of topics, but more verbose than Rudin.
I take Paul Halmos advice when reading math: “the right way to learn mathematics is reading the definitions and theorems and then closing the book and try to discover the proofs for yourself”. It may be overkill at first but you realize it’s a lot more fun that way, and, even if you don’t come up with the right proof, you’ve thought long enough about the pieces involved and it’s easier to see why things are true.
To my money, the best introduction to Analysis in 2023 is Terence Tao’s Analysis 1 & 2, for three reasons: 1) he makes you proof a lot of the main results of the text, a-la Halmos, this makes you really “rediscover” the subject; 2) it teaches you how to approach a math textbook, anyone who has read it know what I mean; and 3) it also teaches you foundations to make the transition more smooth. He literally starts from Peano and takes you through a tour of every number system, and I believe it is something people in 2023 need it given how calculus is taught. And 4! He starts from the real line and then moves on to metric spaces, once the readership can see the point!
This is helpful, thank you!
@@DaMonster you’re welcome! Glad I can help.
The advise on Chunking was very helpful. Thank you for the video. As a side note, I don’t see any one recommending Terrrance Tao’s Analysis book. I am not sure why, but IMHO, it’s one of the best book. It is thoroughly rigorous yet accessible. It takes time to come to the real “real analysis”, it does an excellent job in preparing for the subject matter, by the time you are through volume 2, volume 2 would become absolutely comprehensible
The reason for this is that I don’t have experience with that book. I can’t recommend something I haven’t read. The books I’m suggesting are classical and have stood as standard texts for something like 50 years or more.
Thank you for the references ... I will seek tomorrow for Rudin! Please speak slowly so we could understand ...❤
Nice advice
VISUALIZATION in mathematics is excellent, especially for visual--thinkers! And for self-studiers, answers are essential!
thanks
Turing-Shannon
Great Focus
May Not
End Dissertation
Starts It!
One additions TLA+ from L. Lamport.
Lamport, Shannon, Turing!
Mathematical analysis I and II by Zorich.
Introductory Real Analysis by Kolmogorov and Fomin.
Please do reviews on these books as well.
Just in time! I'm that "math every day" person, btw. I'll be sure to use this when I get to Analysis.
Yeah, when I started studying like that, everything really started to come together for me. I was so baffled nonchalant I was about studying before.
I discovered when I was studying PDE's (which took me down this abstract road) that I would use colors, in PDE you would have the characteristic, general boundary conditons etc and they all got rolled together and it became very tedious to keep track of terms and to establish insight into the various concepts you were blending together. But when you used diverging colors they couldn't hide in the noise of grey pencil marks.
I would also always try to draw a picture of what it was I was representing, again in color to separate ideas (not terms) but ideas, like unit vectors a different color from a regular vector, or is it s apecial eigen vector your using for your solution for a PDE.
I'd say the first step is to familiarize yourself with the basics of how learning actually works. Read "Make It Stick" or "The ABCs of How We Learn" or something like that. The reality of what works is so different from what students and college professors typically expect that there's a lot to be gained from taking a little time for that.
You are nuts! Apostol's Analysis has the hardest exercises ever because it expects you to know things that are not covered in the book or in any calculus book that you've been through! Ever encountered the Vieta's formulas for sums of roots in an ordinary calculus textbook? I don't think so...
Apostol takes more time to go over the details of the proofs than Rudin. It’s a very readable book. Sure, there are hard problems, but there are also a bunch of easier problems. I think the overall number of problems in Apostol are greater than Rudin (don’t have either next to me right now to check though), which gives space for some harder questions like that.
@@JoelRosenfeld I have started Rudin's Principles before a month and I'm already on chapter 3, chugging exercises faster than any of those in Apostol's Analysis... I don't know what are you talking about. Would definitely give Apostol another go and I hope you are right, but as far as my experience goes - Rudin is easier than Apostol... MUCH easier.
Also thanks for the output! Appreciate it.
@@user-eb6mn3dw1v all that matters is that you find a book that works for you. I think Rudin is a masterpiece.
@@JoelRosenfeld flexibility is powerful.
In the latest apostol mathematical analysis, it covers lebesgue integration and complex analysis!
Oh cool! I'll have to check out the newer edition. I was a poor college student when I got my copy. Personally, Apostol's Analytic Number Theory text holds a special place in my heart. I really learned to enjoy Complex Analysis while I was reading that book.
In Europe, we have to take Real Analysis in the first semester of undergraduate, both mathematicians and physicists :(
Measure Theory is also the standard in the second year, and even some physicists take it
It was such a pain in the ass to take Real Analysis, specially as a non-math major
I wish Ive watch this video 2 years ago when I started my major in pure math...
Pughs mathematical analysis book is very picture focused and good.
Rudin book is one shown in thumbnail. The book Baby Rudin is a different one - its written by Maria Silvio. Its available in pdf format. The author is dead.
This might not be a well known book, but Spaces by Tom Lindstrøm is really an incredible book. It completely changed analysis for me. Highly recommended.
Cool! I’ll have to check it out. Thanks for the recommendation!
baffled how real analysis is one of the last courses you take in the Us, while its one of the first(with abstract algebra and linear algebra) math undergrads take here in italy
Yeah, I hear that a lot. The US reshuffles a lot of classes when it comes to math education, but eventually everyone lines up by the time you finish a PhD program. I think the biggest contributors is that in the US, you have to take a lot of general education requirements, and that we don't have very stringent requirements for people entering undergrad.
Hence, many students might have to catch up when they get into undergrad. Some might not have had a calculus course in high school (most math majors have), and they also have to take general classes like Intro to Psychology, English Literature, etc.
That said, there are a lot of undergrads that take Real Analysis in their first two years in the US. These tend to be our stronger students. They usually take Differential Equations and Linear Algebra year one, then jump into Abstract Algebra and Real Analysis year two and three. And after that, they fill up with math electives year four.
The more typical students will coast along and take real analysis at the end. It's a real broad spectrum that we end up seeing. For those students that are ready, they can get a really rich experience in math departments here.
Hi, I'm self-studying real analysis over the summer! I bought the book "Understanding Analysis" by Abbott a few days ago. Do you think this is a good book for self study? Also thank you for this video, it was super useful!
I've heard of Abbott, but it became popular after I finished undergrad. I know a lot of people use it, and it'll certainly get you a decent idea of how things work. Looking through the table of contents, it looks like it only really covers the real line, and like Rosenlicht, doesn't go over Riemann Stiltjes integration.
As for self study, yes, it looks good. Just keep in mind, when you get to a formal class, there might be some extra bits that you are going run into. In fact, it might be better for self study precisely because those bits are left out.
Hey the video is great ...thanks for the tips .... I need to practice more.... could you suggest how I can practice more (any resourses from where I can practice)?
Each text has a good set of problems. That’s honestly the best place to look. Other real analysis problem sets won’t be as helpful, because in a textbook problems correspond to what you have read in the text. Also be sure to carefully work through the examples and theorems in the texts.
The Math Sorcerer :D
I would suggest learning out of Taos analysis books if you’re doing more of a self study type thing. He assumes nothing and the books feel completely cohesive, like the whole thing is one long lecture
@@marigold2257 Tao’s books are good for a self study preparation for a course in analysis. However for a full fledged course, it does have some drawbacks. Many say that his problems are a little too easy, and he puts off the study of metric spaces until the second book.
Rudin does everything much more concisely, but is best read along with an instructor.
If you are looking for another book at about the same level as Tao volume 1, then Abbot’s understanding analysis is a good choice. However, I’d recommend studying Rudin or Apostol before graduate school.
@@JoelRosenfeld I only sort of agree, I think that one should read rudin if they want to learn analysis in a concise albeit painful way but if one wants to enjoy themself while doing it then should read Tao, while yes the problems tend to be easier than rudin, the exposition is better, the problems are more motivated and unlike rudin it’s goal is not simply to teach you what you need to know to get through an analysis course but to also lay the groundwork to better understand more advanced mathematics in the future. Also you say it “puts off metric spaces” but I think this is a dishonest framing, he introduces them later but he covers them in just as much detail as rudin (and also I just like how they are introduced a lot more in Tao compared to rudin) the only real problem Tao has is that the problems are a bit easy which can easily be dealt with by just doing other more difficult problems as a supplement but other than that he covers the same stuff as rudin in the same amount of detail and with the same amount of rigor and I honestly think it’s absurd to say that one has to study rudin even if they have already studied Tao
Amazing video! Thanks so much for all the tips :)
Side question - why not "Understanding Analysis" by Stephen Abbott?
Abbot does a decent job, but it does bypass a lot of material on general metric spaces. You basically learn Analysis only for the reals, but it’s not a big leap to do a more general coverage.
Many people have used and it and liked it. If it works for you, then go for it. Just know it’s missing some things that might come up in later classes.
@@JoelRosenfeld Thanks for the suggestions :)
Rudin-PMA is actually a collection of results of 7 world class text books.
In the name of proofs, only scarce hints are given.
So, if you go through PMA, withdrawal symptom for mathematics will develop in your mind.
That will cause an end of your progress in mathematics.
Remedy:
Simply go through the following seven books in the given order:
1. Naive Set Theory -- Halmos
2. General Topology -- Kelley
3. Finite Dimensional vector space -- Halmos
4. Mathematical Analysis -- Apostol
5. Analysis on Manifolds - Munkre
6. Complex Analysis -- Ahlfors
7. Measure Theory -- Halmos.
Where’s the functional analysis?
@@JoelRosenfeld Now If you want to proceed to learn functional analysis try to master the following two books:
1. Hilbert space -- Halmos
2. Linear Topological spaces. --- Kelley.
@@rajnikantsinha2636 Personally, I’d recommend, Analysis NOW by Pedersen. It’s what I studied when I was in grad school and roughly aligns with Conway’s A Course in Functional Analysis.
I think the ordering in your list is off a bit though. Analysis and Metric Spaces should come before Topology. It helps build intuition for what’s coming and you appreciate the subtleties of Topology more that way.
Halmos’ A Hilbert Space Problem Book should come after a first course in Functional Analysis, in my opinion.
There is a book by Halmos: Hilbert space and spectral multiplicity, I was referring this very book.
If you go through a lot of books on the same topic, your progress will slow down.
Rudin;s Functional Analysis is a good book.
@@JoelRosenfeld
@@rajnikantsinha2636 Ah ok, I thought you meant his other Hilbert space book. Thanks for providing the list! I’m sure many people will find it helpful here. Personally, I’m well past it, as a professor, but I’m happy to find new references.
I’m learning real analysis in my first semester, the exam in the next month
Good luck!
thanks for the recommendation of the Apostol's, which helps me a lot. By the way, I'm learning linear algebra, and can you have some textbook recommendation?
I’m glad it worked out for you. Linear Algebra is a pretty varied subject. If you want to learn it from the axioms up, then I’d say Friedberg, Ingel, and Spence has a good book. There is also Linear Algebra Done Right.
@@JoelRosenfeldThe first one is which I have been using. It’s a big surprise 😊
@@JoelRosenfeldwhat about the abstract algebra? I am going to a next trip 😊
@@Nothinghaigonetochangemylove For a beginner to the subject, I really like Gallian's textbook.
Interestingly, Gallian takes the approach of Groups -> Rings -> Fields, but another standard text is Hungerford's, which goes in the opposite direction. (Hungerford's undergrad book. He has a grad book too)
If you want a tome for abstract, then Dummit and Foote is probably the most comprehensive textbook, but I'd recommend it for a second take on Abstract, after Gallian.
@@JoelRosenfeld Thanks. I will try your recommendation .
is there a nickname for Rudin’s Fourier Analysis on Groups?
Not that I have heard of. Usually I just hear the whole title when people talk about it.
Baby Rudin is best but if anyone wants to study by his own I would say s k mapa real analysis is awesome.
I advise everyone steer clear of The Way of Analysis by Stricartz, my prof insisted on using it and its god awful lol
I don't know how will i pass real analysis is so tough
It is hard. Biggest thing is to carve out a good amount of time to commit to studying every day. Make sure you aren’t taking any other hard classes, and read and read and read. Even Paul Halmos said he had trouble with it in graduate school. You are not alone.
what is r1?
In the United States there are a collection of universities, which are called Doctoral Granting Research Universities. They are essentially categorized at R1, R2, and “Doctoral Granting.” R1 universities are those that have the greatest research output of all the categories.
What ipad apps do you use?
For writing with the Apple Pencil, good notes. If I’m typing up math, then I use Notion which supports mathjax/latex. There is a matlab app that is also really good.
Lol, I"m taking Real Analysis next semester, and I've been scraping by in abstract algebra. Wish me luck I guess
Hello Sir
I am from Kashmir.
Please give I full course on Metric Spaces
as I am not comfortable with Metric Space
There is a whole playlist here dedicated to real analysis, which I think you’ll find helpful
@@JoelRosenfeld
Sir, I need full course of Real Analysis
it really sounds like the real analysis side of maths is barely "maths"
instead of understanding or intuition or anything like that it seemingly is just rote memorization of proof patterns
I wish Idve known this earlier, i treated real anal like i treat other math exams, when i shouldve treated it similiarly to a history exam
It’s not really all rote memorization. But you do need to learn how to use new techniques. This applies to pretty much any math. The best way to do that is to learn how to analyze the examples and theorems in the text.
for most math books from any subject, they keep saying "Proof," but it doesn't prove what they claim all the time. Too many start out with "Assume this is true..." WRONG. Assuming a proof is true =/= proof. You have to PROVE that it's true, so most "proofs" do nothing for me.
On answers: never looking up an answer AFTER you come up with your own is a good way to think you got it right when you didn't. About 50% of the time, the answer I come up with is 100% wrong. I'd never know this if I didn't look it up or use a calculator.
It is important to let yourself dwell on your proof for a while, without looking up another solution. Letting it stew will let a counter example or a weakness in the proof to reveal itself.
Sure, after a couple of days, go look something up after you solved it. But don’t be too hasty. When it comes to research, you don’t have the luxury of having another solution to look up, so you want to train yourself to work without solutions for the simple stuff too.
@@JoelRosenfeld Oh, I'm not used to seeing many proofs show up in the answer key since they consume so many pages. Not sure why there isn't, but there seriously needs to be a class on proof writing before any of the classes that require it show up. But till then, the sloppy method I see in most texts which say they've proven something doesn't help me at all when it comes to creating my own since there are about 50/50 good examples.
@@SequinBrain there are usually intro to proofs classes in mathematics departments. If you want a reference, there is How to Prove it by Polya that is very well regarded.
And for many standard texts, you can find full solutions online. Either as solutions manuals or through places like stack exchange.
@@JoelRosenfeld thx, I'm headed in the direction of proofs, but not there yet. It wasn't in my applied math course degree plan, but I got stopped by topology.
@@JoelRosenfeld Just got the Rosenlicht in the mail, thx! this is way easier to follow and I went back to RA b/c it was one of the classes I didn't understand that well, this should help break through that. good recommend!
Only real intellectuals and true mathematicians use the _Manga Guide to Calculus_ not Rudin pfft
MENTION THAT THESE BOOK LINKS ARE AFFILATE LINKS
I do. In the description. I’m sorry, I’m not trying to deceive anyone
FAIL.
How so? I appreciate any input.
Here’s a good learning tip. Turn off that distracting music in the background.
Ouch. I am sorry you find it distracting. It’s been a process to find the right music. I think my later videos are doing much better with this.
ENJOYING! 😀 A REMINISCENCE OF My MSMath Core Subjects (UP Diliman & CNU, Esp. UPD & Walter Rudin's!..) ONLY We WHO HaD Undergone Such RIGOROUS MENTAL TRAINING!... CAN EVER APPRECIATE ITS GLORY & TRUE VALUE!.. 😀😀😀❤️❤️❤️❤️👍👍👍🇵🇭🇵🇭🇵🇭🇵🇭🦅🦅🦅🦅🦅🙏🙏🙏
Thanks, my goal is to understand Turing's 1936 paper
ON COMPUTABLE NUMBERS, ...
www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
Jay Cumming's book, Real Analysis: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series) 2nd Edition is Great!
Do not forget to check out Shannon's 1938 paper.
www.cs.virginia.edu/~evans/greatworks/shannon38.pdf
mathweb.ucsd.edu/~bdriver/140_F12-S13/Lecture%20Notes/140C_Versions/Math_140C_Ver9.pdf
That looks like a great resource! Thanks for sharing it!