The Real Analysis Survival Guide

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

Yay, a new (for me) math channel with interesting content.

Nice advice

Mathematical analysis I and II by Zorich.
Introductory Real Analysis by Kolmogorov and Fomin.

Please do reviews on these books as well.

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.

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 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!

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.

Your videos are really inspiring.

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.

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.

Turing-Shannon

Great Focus
May Not
End Dissertation
Starts It!

One additions TLA+ from L. Lamport.
Lamport, Shannon, Turing!

The Math Sorcerer :D

Thank you! Suggestions are very helpful.

Pughs mathematical analysis book is very picture focused and good.

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.

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

FAIL.

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

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.

Hello Sir
I am from Kashmir.
Please give I full course on Metric Spaces
as I am not comfortable with Metric Space

Amazing video! Thanks so much for all the tips :)

Side question - why not "Understanding Analysis" by Stephen Abbott?

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

VISUALIZATION in mathematics is excellent, especially for visual--thinkers! And for self-studiers, answers are essential!

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.

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.

Only real intellectuals and true mathematicians use the Manga Guide to Calculus not Rudin pfft

My favourite is introduction to Real Analysis by Bartle and Sherbert

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.

What ipad apps do you use?

Thank you for the references ... I will seek tomorrow for Rudin! Please speak slowly so we could understand ...❤

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

Baby Rudin is best but if anyone wants to study by his own I would say s k mapa real analysis is awesome.

I’m learning real analysis in my first semester, the exam in the next month

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

is there a nickname for Rudin’s Fourier Analysis on Groups?

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 wish Ive watch this video 2 years ago when I started my major in pure math...

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.

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.

what is r1?

thanks

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...

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

How about Abbott understanding analysis

The prof who said do not visualize proofs should be fired . I don't understand where he is coming from 😹😹😹😹

How did students in proof-based math courses get by before proof-writing courses and texts were available?

"Reflexive"? Are you saying that our study sessions should want to become equivalence relations? 🤣🤣🤣

The best book by all means in rigorous one variable, is Understanding Analysis by Stephen Abbott. It has classical proofs (triangle inequality, etc), well motivated and historically intuitive, shows strategies and skrach work of proofs (how to choose an epsilon and when to split it, when contradiction proof is better than direct and vice versa, etc), 10 exercises at the end of each subsections (and not at the end of the chapters), etc. It is very good for self-study and can be read even without an intro to proofs class as Abbott aims also to introduce readers to proofs writing and rigorous intuition (recognizing what in a definition can be used to justify cases). The final chapter introducing more variables cases and metric spaces. And just 300 pages. No solutions, but Abbott explanations are so clear that you can see if you are in the good of bad way ; and as the cases are very classic , every maths forum can tells you if you are wrong or not.

With Abbott you’ll "Understand" Real Analysis.