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Ask HN: Resources to learn real analysis?

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Re: Ask HN: Resources to learn real analysis?

#61

Post by agumonkey »

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 suggested a book that is vastly simpler, so simple it felt like HS but it cleared a few misunderstandings I had about notation and meaning. All of a sudden that 5$ ebay book had more value that my 100$ old paperbrick.

Re: Ask HN: Resources to learn real analysis?

#62

Post by pedrodelfino »

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/...

I have seen "How to Prove It" on Hacker News multiple times! I did a similar question about Discrete Math a month ago [1]. Someone also suggested this book. It is on the top of my "to read list". One of my friends says this book is his favorite one. [1] https://news.ycombinator.com/item?id=16355825

Re: Ask HN: Resources to learn real analysis?

#63

Post by pedrodelfino »

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!

Re: Ask HN: Resources to learn real analysis?

#65

Post by opencl »

mugamma wrote: post #12
throwawaybbqed wrote: post #8

Earlier quoted context omitted.

This one? "Principles of Mathematical Analysis"

Yup. You can get the international version for cheap: https://www.amazon.com/Principles-Mathematical-Analysis-Rudi...

The fact that US version of this book from 1976 remains over $100, while the international version is under $10, and the Kindle version is over 50% more than the paperback, is such a perfect example of everything wrong with the textbook market.

Re: Ask HN: Resources to learn real analysis?

#66

Post by Bahamut »

graycat wrote: post #51
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…

Part II Starting there, as you go on in real analysis , you will place much less emphasis on continuity. The main work will replacing the Riemann integral by something that does much better on the edge or pathological cases. Well, the Riemann integral was, right, a lot about area . Well, it's actually not a really good theory of area -- doesn't have all the nice properties we would wish for. So, that view of area get…

Royden is a classic - my mom and I both used it in our grad math programs, almost 30 years apart or so.

Re: Ask HN: Resources to learn real analysis?

#67

Post by haZard_OS »

Others here have offered some great suggestions already, so I will offer one a little off the beaten path:

Foundations of Mathematics https://www.amazon.com/Foundations-Mathematics-Ian-Stewart/d...

This book is meant to help one transition from performing math in an algorithmic manner to generating proofs based on logic and also set theory.

Re: Ask HN: Resources to learn real analysis?

#68

Post by jinfiesto »

For math majors, baby Rudin is the standard. I would say though that it's probably too terse for most students. I like Pugh's book quite a lot. I think it strikes a good balance between not being long winded while providing enough explanation. Checking the whole answer as you mention is not going to be that helpful in general. There's more than one way to write most proofs and the nature of proof problems is that you have the answer at the beginning. Just make sure you're familiar with basic proof techniques.

Re: Ask HN: Resources to learn real analysis?

#69

Post by yshklarov »

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 the way, as you aquire experience you'll gain confidence and get over the urge to always check your answers. Here's a good exercise with a built-in answer key: When reading a text, every time you get to a result (claim, theorem, etc) try to prove it on your own before you continue. You probably should be doing that more often than not.

In any case, don't stick to just one text/source. Shop around, read a few pages here and there before you settle on something. There's no way a stranger on the internet can make a good recommendation: Find what works for you. The most important thing is that you're fully engrossed!

Re: Ask HN: Resources to learn real analysis?

#70

Post by ghufran_syed »

vaskebjorn wrote: post #54
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? :)…

I can second the Chartrand book (I believe you mean "Mathematical Proofs: A Transition to Advanced Mathematics"). I used this book in a "bridge" class and loved it; it really starts from first principles and gives you a lot of simple examples that you can build on.

Yes, that is the correct title of the Chartrand book, thanks!
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