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Ask HN: Resources to learn real analysis?

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Re: Ask HN: Resources to learn real analysis?

#91

Post by chaboud »

I've spoken with more than one person who made it through Real Analysis intact by reading through "Introductory Real Analysis" by Kolmogorov and Fomin. There's a Dover version that you can probably find for $12 used... It was where I started, but I know several people who found it invaluable after struggling with other texts.

Introductory Real Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486612260/ref=cm_sw_r_cp_apa_a62T...

Come to think of it, there are a lot of good Dover books on mathematics.

Re: Ask HN: Resources to learn real analysis?

#92

Post by kurlberg »

torn_pages wrote: post #85
agumonkey wrote: post #61

Earlier quoted context omitted.

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.

My 5c: both Fraleigh and Dummit & Foote are pretty good.

Re: Ask HN: Resources to learn real analysis?

#93

Post by rocqua »

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…

Really, for me most of my analysis knowledge is based on counter-examples. The counter-examples motivate the theorems and definitions, help check intuitions, and sometimes even help me remember the theorems and definitions.

Re: Ask HN: Resources to learn real analysis?

#94

Post by rocqua »

starchild_3001 wrote: post #84

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 somet…

I have one thing to add. Whenever you meet a new example, check that example against your intuitions. When they mismatch, that is a great learning moment.

These also help in finding 'counter-examples' which tend to be very constructive.

Re: Ask HN: Resources to learn real analysis?

#95

Post by thraway180306 »

starchild_3001 wrote: post #84

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 somet…

This is sometimes called inquiry-based teaching or Moore method, and… there is a textbook advertised as adhering to this particular style, Carol Schumacher's Closer and Closer: Introducing Real Analysis, haven't had it in hand though.

Re: Ask HN: Resources to learn real analysis?

#96

Post by markhkim »

I'm surprised no one mentioned this yet: make sure to talk to your professor, frequently. Introductory analysis courses exist primarily to teach a certain way of thinking, and there is, after all, no better way to learn how to think than to talk to someone who already knows the ways. Take advantage of what you have.

Re: Ask HN: Resources to learn real analysis?

#97

Post by tashi »

pvg wrote: post #89
tashi wrote: post #55

Earlier quoted context omitted.

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

I'm familiar with the expression. Its use here indicates that the reviewer believes the world is divided into those who are smart enough to follow the book and those who are wastes of time.

If it's not obvious why that sucks, imagine going to this professor's office hours if you're having trouble following a proof. Imagine that, rather than thinking you might just not have seen this style of proof before and trying to walk you through it, he indicates you aren't cut out to understand it and suggests you change your major.

(If it sounds like I'm being too harsh based on that one data point, after I read the review I went and looked up the professor on ratemyprofessor.com, and that seems to be his approach to struggling students: that he's weeding out people who just can't learn the material, not people who just got stuck because they're lacking the mathematical sophistication or exposure or confidence they need, or people who are learning more slowly and will be fine if they put in more time and effort and come up with better strategies. Not sure how a professor at Loyola Marymount decided he's qualified to be The Gatekeeper of Mathematics, but there it is.)

Re: Ask HN: Resources to learn real analysis?

#98

Post by agumonkey »

torn_pages wrote: post #85
agumonkey wrote: post #61

Earlier quoted context omitted.

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.

Sorry, first it's a linear algebra, it's by Gareth Williams, linear algebra with applications, I picked the 5th edition iirc, 5$ hardcover.

Re: Ask HN: Resources to learn real analysis?

#100

Post by pvg »

tashi wrote: post #97
pvg wrote: post #89

Earlier quoted context omitted.

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

I'm familiar with the expression. Its use here indicates that the reviewer believes the world is divided into those who are smart enough to follow the book and those who are wastes of time. If it's not obvious why that sucks, imagine going to this professor's office hours if you're having trouble following a proof. Imagine that, rather than thinking you might just not have seen this style of proof before and trying t…

It doesn't really imply that. He just says, a bit stridently, that he thinks the book is targeted at people who will make maths the primary focus of their study. The other stuff is mostly you extrapolating a lot from a single offhanded remark. Reviews are not office hours. Going out to cherrypick additional 'data' doesn't really make your conclusion more sound. The dude wrote some sentence you don't like and next thing you know he's Vinz Clortho the Keymaster of Office Hours.
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