View Full Version : What's a good book for Real Analysis independent studying?

fleshette

03-19-2007, 08:11 PM

Planning to do self study in Real Analysis before I go to grad school in the Fall. Anyone have any favorites? What about user-friendly books?

EDIT: A book with complete solutions (or easy access to a complete manual) is also wanted!

TheBrothersKaramazov

03-19-2007, 08:30 PM

User friendly?? I personally love

Stephan Lay's "Analysis: With an introduction to proof".

This was a nice introduction to future analysis work.

polkaparty

03-19-2007, 08:38 PM

"Principles of Mathematical Analysis" 3rd edition (1974) by Walter Rudin is often the first choice. This book is lovely and elegant, but if you haven't had a couple of Def-Thm-Proof structured courses before, reading Rudin's book may be difficult.

A nice book I recommend for introductory analysis is "Understanding Analysis" (2002) by Stephen Abbott. It covers only typical first semester analysis material (less than Rudin), but it is extremely well written. Every chapter begins with an interesting discussion of problems and examples which motivate the various theorems and definitions.

Another good book similar to Rudin's is "Introduction to Analysis" (1968) by Maxwell Rosenlicht. This book parallels Rudin for the most part, and is slightly easier. Also, it costs only about $10, compared to Rudin: $150 hardcover, $75 soft and Abbott: $40.

econandon

03-19-2007, 08:39 PM

abbott's understanding analysis is good

kartelite

03-19-2007, 09:56 PM

We used Bartle. It was pretty good.

econandon

03-20-2007, 07:04 AM

another good book is Problems and Solutions for Undergraduate Analysis

use it with another analysis book, because it is just lots of problems and solutions. without a teacher to correct your proofs, you want to be able to check your own work. a big part of the whole point of analysis is write proofs well.

reactor

03-20-2007, 09:28 AM

guys, guys, we are looking for a book for self-study. baby rudin, rudin, royden, apostol, taylor etc etc have all very few (if any at all_ worked out examples.

fleshette: order now Mathematical Methods and Models for Economists (http://www.amazon.com/Mathematical-Methods-Models-Economists-Fuente/dp/0521585295/ref=sr_1_1/103-0111986-6539841?ie=UTF8&s=books&qid=1174382452&sr=1-1) by Angel de la Fuente. 1 copy for youserf and 1 copy for your best econ friend. :grad:

(i sent a copy as a gift to my gf. am'I too geeky?) :whistle:

I cannot say enough for this book. Just notice that contains complete solutions (not just out-of-nowhere answers, not just cryptic hints) to all problems in the book! (not to mention that costs less than $50! boy, de la Fuente is a true academic!)

review from amazon:

This an excellent book primarily for students in Economics either PhD or advanced Masters and/or students interested in Economics and who are coming from "rich in maths" fields. It contains a rigorous treatment of the mathematics used in graduate economics. The title "Mathematical Methods" may be somewhat misleading since the book adopts a rather rigorous approach instead of the "cook-book" approach. In many parts the book cannot be distinguished from a pure mathematics book but it includes plenty economic applications and examples.

One of the things I really loved in this book is that it is "self-contained"; at the end of the book you will find complete solutions (not merely answers) of all problems (and not just the odd/even ones). However make no mistake, this is an advanced book and for students with not strong background in maths I would recommend the the sequence (roughly): Alpha C. Chiang "Fundamental Methods of Mathematical Economics" after this, Simon & Blume "Mathematics for Economists" and then de la Fuente's book.

econandon

03-20-2007, 09:43 AM

fleshette: order now Mathematical Methods and Models for Economists (http://www.amazon.com/Mathematical-Methods-Models-Economists-Fuente/dp/0521585295/ref=sr_1_1/103-0111986-6539841?ie=UTF8&s=books&qid=1174382452&sr=1-1) by Angel de la Fuente. 1 copy for youserf and 1 copy for your best econ friend. :grad:

yeah its good but its not a real analysis book, that is a complete treatment of what is typically covered in undergraduate analysis. it has topics covered in analysis but it leaves out a lot. fleshette was requesting an analysis book.

reactor

03-20-2007, 01:47 PM

econandon: you're right. de la Fuente does not cover series or power series for example.

for an introduction:

Fundamentals of Mathematical Analysis by Rod Haggarty (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Rod%20Haggarty)

which has complete solutions to most problems in the back of the book.

for further stuff:

Undergraduate Analysis by Serge Lang (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Serge%20Lang) (there is also available a low-price student-version with chinese cover available from ebay) and its solutions manual: Problems and Solutions for Undergraduate Analysis by Rami Shakarchi (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Rami%20Shakarchi) and Serge Lang (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Serge%20Lang)

JAlfredPrufrock

03-20-2007, 02:06 PM

I liked "The Way of Analysis" By Robert S. Strichartz. He spends alot of time building up intuition for his proofs, although there are some proofs where he kinda waves his hands, and viola, but overall I thought it was usefull.

reactor

03-20-2007, 03:38 PM

there are some proofs where he kinda waves his hands, and viola,

I totally agree on that.

chauchau

03-20-2007, 04:45 PM

Yeah, strichartz is good for an introduction, b/c it's so wordy. I like Browder for a slightly more advanced, and much more concise, treatment.

fleshette

03-22-2007, 05:34 AM

Anymore opinions? I'm only interested in buying one or two books and so far everyone has recommended many different books..

It'd probably be a good idea to ask your math or econ professors, since they're the ones who know either the subject matter or how you're going to need it in grad school.

EconChump

03-22-2007, 03:10 PM

i would not recommend asking professors anything... most of them are so out of touch with reality and how hard it can be to actually learn something new. i think asking people on a forum for advice is precisely the best way to go about it.

i have just ordered abbott and de la fuente. screw the money.

apropos

03-22-2007, 03:24 PM

My undergraduate Analysis course was based on Pugh. I like this book but it's not perfect. The coolest thing about it is that it collects all of the point-set topology concepts into a single, compact (no pun intended) section that's easy to use for review and reference, and it covers somewhat more of point-set topology concepts than others. The section on the function spaces is a very good one too. My course covered only the first four sections, and then I skimmed over the rest of this book. I got a feeling that some of the proofs in the later sections, such as about analysis in R^n section, were too 'slick' and the author takes a bizarre approach towards the Lebesgue integral. He defines it as the measure of a function's undergraph, then derives all the usual properties, then shows than it's equivalent to the usual definition. This was two years ago and now I am reviewing real analysis using Serge Lang's book. I think this text is terrific. Unlike many other texts, it seems like the author is setting you up to study real analysis in R^n from the beginning, and he covers some subtle but important theorems that others don't (for example, did you know that _all_ possible norms in the finite dimensional vector spaces are equivalent to each other or how to Taylor expand a multivariable function?)

phdphd

03-22-2007, 03:33 PM

I recommend Efe Ok's Real Analysis With Economic Aplications. It covers not only analysis in the real line but other topics that will be quite useful for you afterwards (fixed poin theorems, correspondences etc). Besides, the guy really has a good sense of humor, it's funny reading the book. Until recently it was possible to download the chapter's from the author's website but it's not possible anymore (http://homepages.nyu.edu/~eo1/). Anyway, if you want I can send you the whole book. Just PM me.

JAlfredPrufrock

03-22-2007, 04:06 PM

I recommend Efe Ok's Real Analysis With Economic Aplications. It covers not only analysis in the real line but other topics that will be quite useful for you afterwards (fixed poin theorems, correspondences etc). Besides, the guy really has a good sense of humor, it's funny reading the book. Until recently it was possible to download the chapter's from the author's website but it's not possible anymore (http://homepages.nyu.edu/~eo1/ (http://homepages.nyu.edu/%7Eeo1/)). Anyway, if you want I can send you the whole book. Just PM me.

Did he get rid of the Measure theory notes too?

Teazer

03-22-2007, 04:09 PM

although there are some proofs where he kinda waves his hands, and viola......

a musical instrument appears??

Gogol

03-22-2007, 07:32 PM

a musical instrument appears??

That's kind of cheesy, but I laughed :D

Prometheus_Econ

03-23-2007, 01:55 AM

I recommend Efe Ok's Real Analysis With Economic Aplications.

Yes, this is an excellent book because it is rigorous but puts a lot of emphasis on the topics that are really used in economic theory. It's at a higher level than any of the "math camp" books, and I would recommend anyone wanting to do theory to look at it. I'm eagerly awaiting the probability book by him to come out.

Antichron

03-23-2007, 02:53 AM

I actually somewhat enjoyed the book "Fundamental Ideas of Analysis" by Michael Reed. It is written in a much more clear manner than any of my other analysis books. (I have Rudin, Kolmogorov and Fomin, Haaser and Sullivan, and Folland.) Also, Terence Tao (who recently won the field's medal) has a great set of notes available on his websites:

http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/ (http://www.math.ucla.edu/%7Etao/resource/general/131ah.1.03w/)

http://www.math.ucla.edu/~tao/resource/general/131bh.1.03s/ (http://www.math.ucla.edu/%7Etao/resource/general/131bh.1.03s/)

As with most math classes, the best way to learn the material is to do problems. Someone else suggested a book that contains problems and solutions, which could be good. Alternatively, if you want to learn using Reed, my solutions to two quarters worth of problem sets are available on my website at:

http://www.pareto-optimal.com/winter2005/math131a.htm

http://www.pareto-optimal.com/winter2005/math131b.htm

I make no guarantees that any of my solutions are correct, but they are detailed (often pedantically so) and could potentially be of some use. Good luck with analysis!

fleshette

03-23-2007, 04:30 PM

econandon: you're right. de la Fuente does not cover series or power series for example.

for an introduction:

Fundamentals of Mathematical Analysis by Rod Haggarty (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Rod%20Haggarty)

which has complete solutions to most problems in the back of the book.

for further stuff:

Undergraduate Analysis by Serge Lang (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Serge%20Lang) (there is also available a low-price student-version with chinese cover available from ebay) and its solutions manual: Problems and Solutions for Undergraduate Analysis by Rami Shakarchi (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Rami%20Shakarchi) and Serge Lang (http://www.amazon.com/exec/obidos/search-handle-url/103-0111986-6539841?%5Fencoding=UTF8&search-type=ss&index=books&field-author=Serge%20Lang)

Maybe a stupid question: do I need to know this stuff to get through the first year? Can I just read de la Fuente and be good?

I don't want to waste time on what's not important...

Guys, Ok's book is great and he is awesome teaching. At least that is what my classmates agree upon. But as he says it his book is not an introduction to Real Analysis. It somehow assumes you've already taken a serious introduction to the topics and it intends to be an introduction to advanced real analysis. Here at NYU that book is used after Stacchetti covered introduction to Real Analysis at the level of Rudin in his class. So even though I think that the book is really good I wouldn't start there. And just so you know the Probability book is being revised right now as he is currently teaching those topics.

grahamcoxon

03-23-2007, 08:05 PM

I actually somewhat enjoyed the book "Fundamental Ideas of Analysis" by Michael Reed. It is written in a much more clear manner than any of my other analysis books. (I have Rudin, Kolmogorov and Fomin, Haaser and Sullivan, and Folland.) Also, Terence Tao (who recently won the field's medal) has a great set of notes available on his websites:

http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/ (http://www.math.ucla.edu/%7Etao/resource/general/131ah.1.03w/)

http://www.math.ucla.edu/~tao/resource/general/131bh.1.03s/ (http://www.math.ucla.edu/%7Etao/resource/general/131bh.1.03s/)

As with most math classes, the best way to learn the material is to do problems. Someone else suggested a book that contains problems and solutions, which could be good. Alternatively, if you want to learn using Reed, my solutions to two quarters worth of problem sets are available on my website at:

http://www.pareto-optimal.com/winter2005/math131a.htm

http://www.pareto-optimal.com/winter2005/math131b.htm

I make no guarantees that any of my solutions are correct, but they are detailed (often pedantically so) and could potentially be of some use. Good luck with analysis!

Thank you for the notes and the solutions. What's Reed book you are refering to?

EDIT:

As usual, I found the answer following your link (Fundamental Ideas of Analysis). Sorry for the useless post.

grahamcoxon

03-25-2007, 01:35 AM

I am experiencing some problems to find De La Fuente and Reed books (both in phisical and Italian virtual libraries). So I was thinking about spending some happy hours on Rudin in the next months (also because one of my advisor suggested to study on it, actually). But know I am facing a doubt: what is the difference between so-called 'baby rudin' (= Principles of Mathematical Analysis) and 'rudin' (= Real and Complex Analysis)? Which one would you suggest to study on ?

Skipper

03-25-2007, 03:55 PM

But now I am facing a doubt: what is the difference between so-called 'baby rudin' (= Principles of Mathematical Analysis) and 'rudin' (= Real and Complex Analysis)? Which one would you suggest to study on ?I'm reading through "Baby Rudin" right now, and it's plenty challenging for me. I would start with Baby Rudin unless you're a pretty advanced student.

chptlk

03-25-2007, 04:41 PM

I agree entirely with skipper. There are quite a few major theorems that are dealt with in "Baby Rudin"... am reading through it myself.

grahamcoxon

03-25-2007, 04:53 PM

Skipper, I thank you (and chptlk) for your advise. Since I am the proud founder of the club of "math-almost-illiterates who managed to enter an economics phd programme", I think I will spend some time on 'baby rudin' (in my own language, to make things easier) and on some of the math notes available on econphd.net. I hope people on the plane I am taking next thursday won't look at me horrified, as happened to Cassin. ;)

Econ2006

03-25-2007, 04:54 PM

But now I am facing a doubt: what is the difference between so-called 'baby rudin' (= Principles of Mathematical Analysis) and 'rudin' (= Real and Complex Analysis)? Which one would you suggest to study on ?

When people refer to 'Rudin', they usually mean baby Rudin. Rudin's "Real and Complex Analysis" is a graduate-level textbook. You'll probably want to study baby Rudin, unless you've already done several semesters of analysis. Even then, you'll find the problems in that book to be fairly challenging.

reactor

04-08-2007, 06:02 PM

Maybe a stupid question: do I need to know this stuff to get through the first year? Can I just read de la Fuente and be good?

I don't want to waste time on what's not important...

de la Fuente will be more than fine! I say stay simple!

peterB

04-08-2007, 10:59 PM

User friendly?? I personally love

Stephan Lay's "Analysis: With an introduction to proof".

This was a nice introduction to future analysis work.

I second this one. I bought it as a supplement to Rudin because I had very little background in proofs, and it starts at the most basic level. The exercises are helpful and many, incidentally, were on our midterm and final.

concave

09-16-2012, 04:55 AM

"Introduction To Real Analysis", Robert G. Bartle

"

His first book, ''The Elements of Real Analysis'' (1964) a popular mathematics text for graduate students, was published in two editions and translated into several languages, including Spanish, Portuguese and Arabic.

He followed it in 1982 with ''Introduction to Real Analysis,'' which was intended to be accessible to a broader audience, particularly students in related fields like computer science and engineering, said Dr. Donald R. Sherbert, a colleague of Dr. Bartle's at the University of Illinois and a co-author.

Now in its third edition, ''Introduction to Real Analysis'' has sold more than 40,000 copies, a rarity for advanced mathematics texts, Dr. Sherbert said.... "

Robert G. Bartle, 75, Mathematician and Author - NYTimes.com (http://www.nytimes.com/2003/11/03/us/robert-g-bartle-75-mathematician-and-author.html)

concave

09-17-2012, 12:49 AM

Robert G. Bartle, 75, Mathematician and Author - NYTimes.com (http://www.nytimes.com/2003/11/03/us/robert-g-bartle-75-mathematician-and-author.html)

His first book, ''The Elements of Real Analysis'' (1964) a popular mathematics text for graduate students, was published in two editions and translated into several languages, including Spanish, Portuguese and Arabic.

He followed it in 1982 with ''Introduction to Real Analysis,'' which was intended to be accessible to a broader audience, particularly students in related fields like computer science and engineering, said Dr. Donald R. Sherbert, a colleague of Dr. Bartle's at the University of Illinois and a co-author.

Now in its third edition, ''Introduction to Real Analysis'' has sold more than 40,000 copies, a rarity for advanced mathematics texts, Dr. Sherbert said.

JM458

09-17-2012, 03:09 AM

I'm doing an independent study in analysis right now for credit, so I'll give you my take.

Before I start, my background was one RA class with Ross which covered single variable, then one RA class with Browder which covered multi-variable (roughly first 7 chapters of baby Rudin).

Of these, Ross was a decent book and a good introduction, but if you're good at math you may be able to jump into a more advanced book like Baby-Rudin with no background in analysis. I thought Browder was a terrible book with confusing notation and excessively long paragraphs. I didn't grasp this material well until I read the first half of Baby Rudin over the summer, which I thought was a lot clearer.

For my independent study now, I'm using Principles of Real Analysis by Aliprantis and Burkinshaw (Aliprantis was actually an economist with a math doctorate, which is why I picked it). This has a quick review of the sort of stuff you'd see in the first 7 chapters of Baby Rudin, and then a chapter generalizing results from metric spaces into a general topological space. This was my first introduction to Topology and I thought it did a reasonable job. Starting with chapter 3, it gets into more advanced material like measure theory, lebesgue integration, hilbert/banach spaces.

Overall, I really like this book so far. It's well written and the exercises are challenging but not excessively so. The authors also published a book called Problems in Real Analysis which has the solutions to the hundreds of exercises in the book, which you can find online. If you're self studying, a solution manual is very helpful so you can check whether your solving the exercises correctly.

Edit: If you are self studying, there is a video series on YouTube that goes through the first half of Baby-Rudin at Harvey Mudd College, and the teacher is very good. This is definitely how I would self study if you are a beginner, just make sure you do some exercises.

Also, I found Ok's book to be a little confusing, definitely don't use it as an introduction.

kennysmith

09-20-2012, 07:27 PM

Abbott is absolutely the best, if you've never even seen, e.g. how to prove that sqrt(2) is irrational (or even if you've seen that and a little more), and are planning to study analysis on your own.

You need to be pretty bright to be able to do Baby Rudin on your own, if you've never taken a class involving proving stuff before. He is famously terse, even in that "Baby" book.

Also, you may or may not be able to find a copy of Abbott to briefly peruse, quite easily on a certain Swedish website.

Danielle Burgo

11-25-2012, 09:44 PM

thank you!