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Ask HN: Resources to learn real analysis?

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Re: Ask HN: Resources to learn real analysis?

#41

Post by jfv »

defen wrote: post #34
ghufran_syed wrote: post #29

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

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

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

Re: Ask HN: Resources to learn real analysis?

#43

Post by x0x0 »

jfv wrote: post #39

Do NOT read Rudin. He is terse and unless you are already well versed in mathematics it is simply incomprehensible. My recommendation would be Spivak’s calculus. There are a million great exercises and the book is beautifully typeset and overall a pleasure to read. Don’t let the title fool you, there are analysis exercises in there.

You have to decide between two things:

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

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

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

Re: Ask HN: Resources to learn real analysis?

#44

Post by totalZero »

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

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

https://ocw.mit.edu/courses/mathematics/18-100a-introduction...

https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-f...

https://ocw.mit.edu/courses/mathematics/18-100c-real-analysi...

Re: Ask HN: Resources to learn real analysis?

#45

Post by spacedome »

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

Re: Ask HN: Resources to learn real analysis?

#46

Post by [deleted] »

thraway180306 wrote: post #16
VideoEveryDay wrote: post #4

Rudin's classic texts are a great resource.

Rudin is a good breviary rehearsal if one already almost-knows and feels the material. People with certain inclinations may get a warm fuzzy feeling how things neatly fit together as if by omniscient design. Otherwise it is a crossword puzzle to amuse the god himself. You may get a feeling you understood things (and earned that), but you are wrong. When I ask people what they really remember Rudin from, what specific…

[deleted]

Re: Ask HN: Resources to learn real analysis?

#47

Post by AlexCoventry »

jfv wrote: post #39

Do NOT read Rudin. He is terse and unless you are already well versed in mathematics it is simply incomprehensible. My recommendation would be Spivak’s calculus. There are a million great exercises and the book is beautifully typeset and overall a pleasure to read. Don’t let the title fool you, there are analysis exercises in there.

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

At the very least, it has great exercises.

I also loved Spivak.

Re: Ask HN: Resources to learn real analysis?

#48

Post by edanm »

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

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

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

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

Re: Ask HN: Resources to learn real analysis?

#49

Post by thraway180306 »

auntienomen wrote: post #32

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

Also Shilov.

Re: Ask HN: Resources to learn real analysis?

#50

Post by opencl »

ghufran_syed wrote: post #29

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

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

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

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