Real analysis self-study book

May 13, 2008 9:50 PM

What's a good real analysis textbook for a self-learner who doesn't have much of a pure math background?

I've just finished a degree in electrical engineering; what better way to celebrate than to spend the summer studying real analysis? I'll be starting a master's program in mathematical finance in fall; by all accounts, the program is very fast-paced and a lot of work, so I want to be up to speed on everything before I start.

As an EE, I never really saw a lot of "real math"; I did your standard Calculus I-III sequence, linear algebra, differential equations, etc. but those classes were mostly computation. This semester, I took a more theoretical course in partial differential equations and had a hard time just reading the textbook, let alone doing proofs. I think the math in my masters program will be more applied than theoretical, but they still do recommend students to have "general knowledge of real analysis, measure and integration, ..."

I've got three months until the program starts and not a whole lot to do, so what analysis books would you recommend for someone who doesn't have any pure math background? I'd like something that goes slow and holds your hand in explaining all the proofs; it's hard for me to follow really terse textbooks where the author assumes that what he's doing is obvious. I know people swear by Rudin, but I honestly don't think I have the background for it.

posted by pravit to education (12 answers total)

Principles of Mathematical Analysis is a classic, it was my text book when I took analysis.
posted by mto at 10:03 PM on May 13, 2008

mto, he doesn't think he has the background for Rudin.

Like most former math majors, I used Rudin in my analysis classes; like many former math majors, Rudin was waaaay over my head. Don't even bother. If I were you, I'd be looking for books called "Advanced Calculus" rather than "Real Analysis". They could be substituted for each other in my major program but Advanced Calculus was the class for people who were sure they didn't want to continue on to grad school in math. I assume it'll be more concrete, while giving you a little more theoretical background than you got from your calculus classes.

I can't recommend any specific books, unfortunately. I think your instinct to stay far from Rudin is a good one; I know people in their fifties who still resent that text.
posted by crinklebat at 10:52 PM on May 13, 2008

Are you sure you need real analysis for a mathematical finance program? Seems impractical.

I needed it for my Mathematics and Computer Science major.. which is a little more believable from a theoretical standpoint, but honestly it still never ended up being practical / useful in my career life.

That stuff made my head explode. I just took a look around my place to try and find what book to tell you NOT to use, but I couldn't dig it up. It was an old, red book with gold-foil-ish lettering and I think it was simply called "Real Analysis." It sucked verily.

Ahh, I've found it on Amazon: here you go...

I guess maybe it'd work better for you than it did for me, as I never ended up reading a Real Analysis book again in my life.. and couldn't tell you if this was better/worse than others.. all I know is the class drove me freaking insane.... until I got to complex analysis at which point I totally flipped out like a ninja and committed to turning my back on mathematics forever. Kinda.
posted by twiggy at 11:10 PM on May 13, 2008

Despite your concerns otherwise, I would still suggest Rudin's book for this purpose. It covers exactly the material you would like and, in all honesty, it's probably one of the easier to read expositions into the material. If you're willing to work at it, and do the exercises, then you should be fine.
posted by vernondalhart at 12:39 AM on May 14, 2008

I'm kind of inclined to suggest you go to a strong calculus book and revisit what you know with more depth, paying attention to foundations and proofs of theorems and the "large storyline" more than the computations this time. Apostol's Calculus is a good choice for this, but I particularly recommend Spivak's (equally creatively-titled) Calculus. You can learn to Think Like A Real Analyst from those, and in your position it sounds a better choice than something as compact as Rudin. There is a "supplement" to Spivak which has solutions, including proofs.

The biggest difficulty about doing this alone is that real analysis is often about (1) writing clear and sound arguments, and (2) not missing small subtle easily-overlooked points. It's very difficult to check yourself on those things. If it's at all possible you can get someone to work with from time to time to let you know when you've missed a small point (or when you're totally missing the point), it'd be well to your advantage.
posted by Wolfdog at 3:35 AM on May 14, 2008

I don't think you'll get a lot out of a heavy duty real analysis book like Rudin, and I don't think it will be that helpful for mathematical finance anyway (beyond giving you some of the right habits of mind for that kind of mathematics).

For a halfway house (real proofs but fairly expository and covering various ideas in analysis and integration) you might like Tom Koerner's Fourier Analysis book. He's also written A Companion to Analysis (used to be free download on his website but no more) but I don't know much about that one.

The other thing I'd recommend (if you haven't already) is to get a good quantative finance book. Hull is standard, but if you want exposition of the ideas before you tackle that then I'd suggest Wilmott Introduces Quantative Finance. If you read that you'll have a lot better idea where your mathematical weak spots are.
posted by crocomancer at 3:36 AM on May 14, 2008

I used Patrick Fitzpatrick's _Advanced Calculus_ for my undergraduate text, and my recollection was that it was reasonably accessible. (I haven't looked at it with the eyes of a professor, because I don't teach analysis, but my recollection is my colleague who does teach analysis thought it was pretty good, too.)

Don't use Baby Rudin. Even as a graduate student in mathematics, it was pretty terse and opaque.
posted by leahwrenn at 4:56 AM on May 14, 2008

I'll recommend another one, Lay's Analysis with an Introduction to Proof precisely because it does spend real time talking about the structures and styles of proofs, which I think is likely to be helpful. You might also consider picking up a book specifically about proofs, like Solow as a complement to whatever you settle on as your main book on analysis.
posted by Wolfdog at 5:04 AM on May 14, 2008

Counterexamples in Analysis is a good book to have around regardless of what text you're using, and it's only $10.

R.P. Burn's Numbers and Functions: Steps into Analysis seems ideal for self study. I haven't used it myself, but I read his Number Theory book, which takes the same approach - it has worked solutions for all the problems, and has problems interspersed throughout the text, forcing you to work through all the major results.

I've heard Apostol's Mathematical analysis explains things in more detail than Rudin, but I haven't read that either; might be something to look into if you want a more traditional approach.

Personally, I really liked Browder's Mathematical Analysis: An Introduction, but a lot of amazon reviewers pan it as being too dense.
posted by suncoursing at 2:27 PM on May 14, 2008

Thanks for your suggestions, everyone. I still haven't decided on a book, but for posterity, here's some books others recommended:

Elementary Analysis: The Theory of Calculus by Kenneth Ross

Yet Another Introduction to Analysis by Victor Bryant

A Friendly Introduction to Analysis by Witold Kosmala

I'm quite tempted to get Bryant's book, but we'll see. Anyone here have experience with the books I mentioned above?
posted by pravit at 2:30 PM on May 14, 2008

On the basis of crocomancer's answer, I went to Korner's website to mourn that I was too late to download his Companion to Analysis, I guess-- his Fourier Analysis is a fabulous book-- and I found some free downloads there that might interest you, such as:

New! A second edition (even better than the first) of an advice sheet by Mark Joshi for Maths PhDs thinking of going into finance (he says add `suitable disclaimers blaming him for anything in it' so I do): PDF

The "advice sheet" is 16 pages long and includes reading recommendations (Wilmott, Hull, and others) and addresses of fora where people discuss books and job prospects.
posted by jamjam at 6:00 PM on May 14, 2008

Oh BTW, just as a followup, I ended up getting Yet Another Introduction to Analysis. It seems pretty good so far, although I haven't gone through much of it yet.

I plan on going through Capinski and Kopp's Measure, Integral, and Probability when I'm done with this one, if I have time. It looks like it would have more direct application to mathematical finance, but I'd like to have a solid base in analysis first.
posted by pravit at 8:01 PM on June 4, 2008