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Ask HN: Resources to learn real analysis?

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Re: Ask HN: Resources to learn real analysis?

#71

Post by thraway180306 »

pedrodelfino wrote: post #63
fofoni wrote: post #56

Hi, 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 b…

Thanks, 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!

Law+Math degree is definitely on the weirder side of things I've heard (and I endorsed Oxford's joint CS and Philosophy). But in that case I may allow myself to make an equally far stretch connecting something lawyerly with real analysis (that isn't about RA utility in data science for law enforcement). About how model theory shapes logic. It is advanced (practically algebraic geometry now) and not really related any more, but the basic issue came from set theory and analysis: models of infinitesimals, like in Keisler books. Since then it came to encompass and classify all logic-based mathematics (not every creative reasoning in mathematics is logical! though papers always are) and more exotic logics such as the „default logic” sometimes employed by lawyers.

It's Bressoud BTW, I endorse that too. Along with TW Körner „Companion to Analysis: A First Second and Second First Course” with Lang or Zorich as base. I wouldn't be as insistent on pencil at all times if it were to prevent broader reading or just expanded skimming.

Re: Ask HN: Resources to learn real analysis?

#72

Post by adyavanapalli »

fofoni wrote: post #56

Hi, 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 b…

To piggy back on this, Devlin has an excellent free course on Coursera roughly following the textbook.

Re: Ask HN: Resources to learn real analysis?

#73

Post by marktangotango »

trendia wrote: post #36

In 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 fu…

When 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.

Re: Ask HN: Resources to learn real analysis?

#75

Post by westoncb »

yshklarov wrote: post #69

I 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 t…

I'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.

Re: Ask HN: Resources to learn real analysis?

#76

Post by llamaz »

westoncb wrote: post #75
yshklarov wrote: post #69

I 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 t…

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

Re: Ask HN: Resources to learn real analysis?

#77

Post by llamaz »

yshklarov wrote: post #69

I 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 t…

Rudin is a good reference book. It's not good to learn from, but if you're looking for the authoritatively _best_ proof, that cuts right to the heart of the problem, then Rudin is great.

It's not good for self study, but I used it as a supplement to a real analysis course I was taking at the time, and I appreciated the terseness and dryness. My lecturer went into the details, so it was good to have a terse book to remind me of lecture content when doing assignments and studying for exams.

Re: Ask HN: Resources to learn real analysis?

#78

Post by internetman55 »

graycat wrote: post #40

Part 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 for…

Thanks for your in-depth posts!

Re: Ask HN: Resources to learn real analysis?

#79

Post by westoncb »

llamaz wrote: post #76
westoncb wrote: post #75

Earlier quoted context omitted.

I'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'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?

Re: Ask HN: Resources to learn real analysis?

#80

Post by tzs »

opencl wrote: post #50
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…

I think we make this harder than it has to be, by not doing proof-based math earlier.

Real analysis is often the first class in a math bachelor's program where things really get far from intuition and much harder to wrap your head around than the material from earlier classes.

Making that the first proof-based class is just piling one hard thing on top of another.

I think we'd be better off if we made basic calculus proof-based, at least for people such as math majors who are going to need to learn to follow and do proof-based math at some point. For those going into fields where they will not need to read and do proves, have a separate "practical calculus" track.

You can start out with more informal proofs at the start of basic calculus, and slowly step up the level of rigor throughout the year.

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