Real Analysis Textbooks

[FierceEconDR]

Originally Posted by greeneggs

Another of my favorites is "A Radical Approach to Real Analysis" by David Bressoud. It's really a great book, it tackles Real Analysis in a completely different way, basically taking you through it chronologically as it was discovered/developed; and providing you with the history and intuition - as well as mistakes great mathematicians made - necessary for you to really appreciate what you're doing and why you're doing it that way, and what other ways you could have possibly done it. That's something I think is lacking in pretty much all other books, especially definition-theorem books like Rudin.

That is the best book ever! Readable and the rest you can get from Baby Rudin! And if you are willing to use/learn some maple/mathematica the exercises are without comparison!

[SubEffect]

I used the Rosenlicht book and thought that it was great (and cheap!). If you're feeling nervous about proof based math, then there are a couple of books that give good introductions to how to write proofs, and the logic behind it. "How to Prove It" by Velleman is pretty good, and I would recommend it to anyone in that boat.

[corolla09]

Elementary Analysis by Kenneth Ross,

May be a little basic for those who took Real Analysis already but those who have not it is a good text to teach yourself a little before jumping into Math Camp this fall. Or a good introduction before reading Rudin. If you didn't do any delta, epsilon proofs in your calculus classes I would suggest this text first. It is short and easy to read but has all the proofs needed (at least for a pre-req for econ this fall).

[Econtastic]

Originally Posted by corolla09

Elementary Analysis by Kenneth Ross,

May be a little basic for those who took Real Analysis already but those who have not it is a good text to teach yourself a little before jumping into Math Camp this fall. Or a good introduction before reading Rudin. If you didn't do any delta, epsilon proofs in your calculus classes I would suggest this text first. It is short and easy to read but has all the proofs needed (at least for a pre-req for econ this fall).

Agreed, I thought this book was pretty solid for an introduction to analysis. Of course they love it here at Oregon, because Ken Ross used to teach here

[BenedictProsper]

Is real analysis really that important? I don't know how much I should learn about real analysis? What role does it play in grad courses?

[greeneggs]

Originally Posted by BenedictProsper

Is real analysis really that important? I don't know how much I should learn about real analysis? What role does it play in grad courses?

it's important because of the skills you develop. It isn't useful in the sense that there's awesome theorems or anything that solve or make your problems easy. It's important in the vein that the things that you learn in introductory Real Analysis are very basic and fundamental to even getting started on a problem in most cases. It's basically a review of calc 1 and calc 2, but this time focusing on what you're doing, why it works, and what assumptions you have. It basically teaches you the proper way to prove things.

Graduate Real Analysis is the same thing; except you actually do get a bunch of awesome theorems that make solving your problems trivial. We actually used a lot more grad real analysis than undergrad in our grad Micro class; the only thing is that we just took it for granted and didn't prove the more complicated theorems (some which were homework assignments in the grad Real Analysis class). That said the grad Real Analysis class was probably 5x the work and 3x as hard as grad micro; so maybe it isn't worth it if you're busy. You could go through Royden or Folland; but I would just recommend buying "Infinite Dimensional Analysis: A Hitchhikers Guide" by Aliprantis and Border and keeping it around as a reference. It has all the good theorems and everything you'd probably ever need analysis wise; and it's written by a guy who's interested in Math Econ so it is tailored toward stuff you may actually use.

[Gecko]

Efe Ok's book, Real Analysis with Economic Applications is very user friendlytoo

[Invisible Elbow]

Originally Posted by philecon

You may want to look at Understanding Analysis by Stephen Abbott.

I highly recommend this book. It is very much more introductory than some of the other texts mentioned in this thread, and it does not ever get to a very high level because it deals only with analysis of the real line. But it is very good at imparting what it does cover; its major strengths are Abbott's exposition, which is unusually clear, and the motivation he gives for doing what he's doing. There is also some introduction to proofs along the way. Conversely, books like Rudin or Kolmogorov, which are little more than collections of theorems, may leave you mystified not only about the concepts, but also about why you're doing this "analysis" nonsense at all.

We're using Kolmogorov in my analysis course. It seems like it might make a good reference, but there are lots of lines like "the proof is familiar from elementary analysis" and "the reader is likely already familiar with the Heine-Borel theorem from elementary analysis" —I read that, shook my head, and went out and bought Abbott's book. It's been a great aid in righting myself after being flattened by Kolmogorov's broadside.

And now I am going to buy Bressoud's Radical Approach on strength of the recommendations in this thread.

[moneyandcredit]

I'm currently reading through "Introduction to Real Analysis" by Michael Schramm in preparation for taking Intro to Mathematical Analysis next semester. It ain't Rudin, or even Rosenlicht, but it looks like a good way to bridge my way up to them.

[invain]

At an introductory level:
Terence Tao, one of the greatest mathematicians among the new generation, has collected lecture notes for Honors Real Analysis classes in Analysis I-II, well I haven't used myself and examined the books in detail: excellent flow of material, starts from abc's of analysis up to advanced topics like measure theory and Lebesgue Integral -OK, measure theory may not be that advanced since almost every graduate analysis course begins lectures with measure theory but it is "advanced" for a beginner in analysis- and for people who are at a good level of math, I'd say it is a good reference book if you need one. Nevertheless, Tao himself follows Rudin in his classes I suppose

Efe Ok's book more than excellent...but I don't suggest it for someone studying analysis as a preparation to econ phd programs. If you have taken a real analysis sequence before but have no idea about mathematical economics (in a very general sense - not implying a field of economics), it is a fascinating book to review analysis while explicitly working on applications to problems in economic theory and believe me it will help a lot if you are going to do research on economic theory. but like I've said, if you are looking for a book to study analysis yourself before mathcamp or whatever, go with Tao