Baby Rudin necessary?

osnarf

I was preparing to read through this thoroughly from all I've heard about it, but the topics seem to be very similar to Spivak's Calculus, that I just worked through, and Spivak's Calculus on Manifolds, which I'm currently working through.

I was wondering if there is any compelling reason to read the whole book (different or more in depth treatments of subjects), or if I should just read through the sections that weren't covered in Spivak's books (Fourier Series, Lebesque integration, basic topology, etc) and move onto something more advanced in the same subject area, such as Rudin's Real and Complex Analysis.

Any thoughts? If you go with the latter, any suggestions are appreciated.

qspeechc

I am not familiar with Calculus on Manifolds, but I just wanted to say that baby Rudin does not treat Fourier series, Lebesgue integration and topology very well, and you will have to learn these subjects elsewhere anyway.

mathwonk

Although friends of mine who were analysts swore by Rudin's books as the best, I have never liked them. I prefer Lang or Cartan for complex analysis, and Berberian for real analysis.

deluks917

I don't think its worth reading at all. I'm currently reading Serge Lang's Real and Functional Analysis with just Spivak and some Algebra as Background. While I find the book pretty difficult I feel like my background isn't a problem. I'd read Rudin's real and Complex Analysis or Maybe Folland.

Also look at MIT's Undergraduate analysis courses analysis 1 and 2. The first course uses Rudin POMA and the second semester uses Spivak Calculus on Manifolds. I think after Spivak you will be very well prepared for graduate real analysis. One thing that does help is a little bit of algebra, for example most books discuss ideals of continuous functions on compact sets.

snipez90

Obviously only read the sections you're weak on unless you're not confident with your basic analysis skills (e.g. if you don't know how to do all of the problems in Spivak, it's probably worth revisiting the relevant topic). However, I think Rudin's coverage of topology (chapter 2) is worth reading through since unsurprisingly, it's oriented towards analysts. IIRC, Spivak basically talked a bit about compactness in the manifolds text and that was it. In particular, solely reading Spivak will not expose you to metric spaces, which is like the MOST important concept in undergrad analysis.

Also, Rudin PMA chapter 7 on sequences and series of functions is something you won't find in Spivak. Rudin's exposition is cleaner and he goes further in depth on uniform convergence than baby Spivak (partly thanks to metric topology). Also the Stone-Weierstrass theorem is just cool.

The reason you shouldn't read the Fourier series and Lebesgue integration sections are not because they're not covered well. The Lebesgue integration stuff overlaps with the first chapter of Real and Complex Analysis. On the other hand, proper treatment of Fourier analysis really depends on tools from measure theory and Lebesgue integration.

osnarf

Phew, because I didn't really want to :) I felt confident with the information in Spivak so i'll just read through chapters 1 and 7 and then move on (I just picked up Mendelson's topology book, so I'll skip over Rudin's chapter in it).

I've heard good things about Lang, never heard of Berenian or Cartan before, I'll look into them. I find it interesting you (snipez90) say that about Fourier series, because the Stein and Shakarchi book series I was look at goes:

Fourier Analysis
Complex Analysis
Real Analysis

I also find it interesting that they put complex analysis before real. Is that normal, or is a different order usual recommended? Also, have any of you read this series?

mathwonk

Everything written by Henri Cartan is beautifully, clearly and elegantly written. Everything written by Sterling K. Berberian is extremely clearly written and explained. Lang is a little spotty, clarity varying from book to book, but I like his complex analysis book.

Integral

Frankly, I must object to the terminology "reading" , you do not "read" a math text, you must work your way through it. If you are only going to read it, save your time, get a good novel.

osnarf

Mathwonk: It's funny, because I was going to ask you which book by Berberian you meant, until I was reading reviews on amazon and saw that Fundamentals of Real Analysis by Berberian had just one review, written about 3 1/2 weeks ago, by a familiar name :P I'm assuming that's the one you were referring to? Also, in reference to this post, are Lang and Cartan's book in the category that you recommended reading after real analysis to get the full benefit, or would it be fine to read them concurrently?

Integral: I am obviously planning on working through whatever i do read. I have pen and paper in hand the entire time and i do almost all the exercises in books in math/science subjects. To me, it just sounds more natural to say I am reading a book than working through it. Like you said, it's terminology.

mathwonk

berberian's "fundamentals..." is excellent and i have it here now. but way back in college i also read his book on hilbert space and it too was superbly clear, so much so it seemed almost like cheating to read it as he explained everything understandably. i wondered why other authors did not do so. reading berberian makes one wonder if other authors are on an ego trip to make you think they are smart rather than just help you out.

so i conclude all books he writes are probably the same, super clear.

as to lang and cartan on complex analysis, you need basic elementary limits and so on (such as available from spivak's calculus), but not lebesgue integration (as in berberians fundamentals... ) before reading them.

ideally go hang out a couple hours in a university library and browse these guys to get your own take on them.

osnarf

I was actually asking between fundamental's and A First Course, but after looking a little closer the former was what I was looking for, so I'm going to check out that, and Cartan for complex. The hilbert space book sounds interesting too, but i'll probably hold off on that for a while.

I'll probably wind up reading either Cartan first or both at simultaneously, because I have been waiting since early high school to find why in the world we care about these imaginary numbers :)

Thanks again everyone, you were very helpful as always.

Landau

I would not work through 'baby Rudin', certainly not the last chapters, they are treated better in other sources. But there is still a gap between Spivak's two books and Rudin's Real and Complex Analysis (the latter is great by the way, as is his Functional Analysis). This gap consists mainly of topology in the context of metric spaces, as Spivak does everything in R (Calculus) or R^n (Calculus on Manifolds).

I think one of the best things you could do, is read Knapp's Basic Real Analysis. Knapp has become one of my favorite authors, I really really like his style. He excels in clarity, and makes original choices of the order presentation. If you do his Basic and Advanced Real Analysis, you know everything a graduate student needs to know about analysis. Similarly for his Basic and Advanced Algebra. Indeed, that was his goal: to cover everything a mathematician, not necessarily specializing in that area, is expected to know.

Other good books are Apostol's Mathematical Analysis and, somewhat more difficult, Dieudonné's Foundations of Modern Analysis. The latter contains an extensive treatment of metric spaces.

mathwonk

dieudonne's quote in chapter 8 about "slavish subservience to the shibboleth of numerical interpretation at any cost" is worth the price of admission alone.

Landau

Right, and

Fredrik

These books look really interesting, since the table of contents mentions a lot of things that I would like to know more about. I just have to say that it seems really weird to use the term "real analysis" in the title. Why not just "analysis"? I wouldn't even have looked inside it if you hadn't recommended it, because the title would have made me think it's just an alternative to baby Rudin.

Landau

Because it is mainly about analysis in R^n, not e.g. complex analysis. A lot of courses have the name 'real analysis' when they are about measure and integration, even functional analysis. (I have learned not to pay too much attention to artificial names that people happen to use.)

osnarf

Would you say the scope of the basic Knapp book you described is similar to the fundamentals book by Berberian mentioned earlier? I looked at the tables of contents and they seem similar, but it's all Greek to me (for now).