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Ask HN: Resources to learn real analysis?

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Re: Ask HN: Resources to learn real analysis?

#81

Post by mlevental »

tashi wrote: post #55
jhck wrote: post #9

Tao's Analysis I is fantastic. There's a review available from the MAA: 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 presid…

Holy 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."

and people wonder why mathematics is reviled. talk about elitest.

Re: Ask HN: Resources to learn real analysis?

#82

Post by llamaz »

westoncb wrote: post #79
llamaz wrote: post #76

Earlier quoted context omitted.

That's weird - I think Rudin is great and do everything through a terminal through emacs...

Haha—that's great! But wait, you weren't in Berkeley, CA in 2013 were you?

I'm from Australia. My theory is that of the subset of people that are interested in both computer science and math, a significant portion use linux and if you use linux, then emacs is the best LaTeX editor (auctex and reftex are amazing). And "doing everything in emacs" is just what naturally happens when you use emacs long enough.

Re: Ask HN: Resources to learn real analysis?

#83

Post by westoncb »

llamaz wrote: post #82
westoncb wrote: post #79

Earlier quoted context omitted.

Haha—that's great! But wait, you weren't in Berkeley, CA in 2013 were you?

I'm from Australia. My theory is that of the subset of people that are interested in both computer science and math, a significant portion use linux and if you use linux, then emacs is the best LaTeX editor (auctex and reftex are amazing). And "doing everything in emacs" is just what naturally happens when you use emacs long enough.

Makes sense. And so did your explanation for liking Rudin above (or wherever it's positioned now).

Re: Ask HN: Resources to learn real analysis?

#84

Post by starchild_3001 »

Ha! 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.

Re: Ask HN: Resources to learn real analysis?

#85

Post by torn_pages »

agumonkey wrote: post #61
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? :)…

One personal advice: if you want to learn a subject, don't focus on one book, one author or one set of suggestion. Only care about what makes your mind walk the domain. 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 sugge…

Which abstract algebra book if you don't mind me asking.

Re: Ask HN: Resources to learn real analysis?

#86

Post by auvrw »

your 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.

Re: Ask HN: Resources to learn real analysis?

#87

Post by meuk »

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? :)…

This is so true! I have found many textbooks very hard to work through. One you somewhat master the subject, the book seems exceedingly clear and you wonder why you ever struggled.

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.

Re: Ask HN: Resources to learn real analysis?

#88

Post by MaysonL »

You 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.

Re: Ask HN: Resources to learn real analysis?

#89

Post by pvg »

tashi wrote: post #55
jhck wrote: post #9

Tao's Analysis I is fantastic. There's a review available from the MAA: 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 presid…

Holy 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."

It’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

Re: Ask HN: Resources to learn real analysis?

#90

Post by Jugurtha »

May 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
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