What's to come after little rudin?

[jonasK]

So I'm sort of done with little rudin. I've got an amazon gift certificate and I want to spend it on some worthwhile books. I don't know enough algebra save for some fields theory, but I'm more inclined to do analysis. What's next? measure theory? topology? I'm open to suggestions.

 

[HallsofIvy]

Big Rudin? What's next is entirely up to you. If you have mastered the basics of mathematical analysis then you probably already know enough basic topology to start on "functional analysis" and "measure theory" but it wouldn't hurt to get a little general topology under your vest.

 

[phreak]

How about H.L. Royden's Real Analysis? It's "baby" graduate analysis, starting out with (Lebesgue) measure theory and functional analysis, and then branching into more complex topics such as abstract spaces.

 

[quasar987]

u said u don't know algebra. why not go for an algebra book? it's a beautiful subject and has lots to teach you about becoming better at math altogether in terms of how to think.

'Algebra' by M. Artin

 

[jonasK]

Thanks all. I ordered Big Rudin! I also found Royden's Real Analysis and Artin's Algebra in the library, so I'll definitely check them out. As for Topology, I'll look for something. Oh, and if anyone has recommendations beyond analysis, please chime in too!

 

[therector24]

may be you can start with thomas calculus book, i think it will help

 

[SiddharthM]

munkres is a good topology text. complex analysis at the level of ahlfors or conway also seems like a good next step. I'm also wondering what I should be doing next, I may go ahead and begin with the next Rudin but I've used Royden before and thought it was well written.

royden vs. big rudin, pros and cons?

 

[rudinreader]

I also recently finished 1-7 of Baby Rudin. I'm about to start reading Conway's Complex analysis 1 book, and also hopefully reading Adult Rudin simultaneously. I would rather just read Adult Rudin, but the qualifying exam next fall is going to be based on Conway.

I have glanced at the contents of Royden, and the con's are that there's too much time spent on Topology, given that Baby Rudin already cover's a lot, such as some nontrivial facts like a metric space is compact if and only if every infinity subset has a limit point in it.

The pro's of Rudin, in the words of one of my math professor, is that most Phd math students don't learn so much as is in that one book in their entire Phd studies! The con's is that it's an intimidating book. But recently I realized that most of the problems in chapter one are actually easy, with the exception of #1 - proving that there is no countable infinite sigma algebra. That one is hard.

The other books I am aware of, that are introductory (yet they are still hard!) and cover significant ground, are Bredon's Topology and Geometry, and Miller's Symmetry Groups (out of print, but available online at: http://www.ima.umn.edu/~miller/symmetrygroups.html ).

But the more people reading Conway and Adult Rudin, the better. That's what I'm reading, and can discuss problems online. I'm thinking about reading Topology and Geometry, etc. down the road.

As for algebra and differential equations, those subjects are side effects. Analysis is the basic prerequisite for Math that treats Physics as a whole.. which is what I'm interested in, and it a source of examples in Topology/Algebra, etc.

 

[phreak]

I think big Rudin is meant for a more mature reader, and I think going from little Rudin to big Rudin can be quite a big step (but definitely doable if the reader has the patience).

Here at the University of Chicago, Royden's Real Analysis is used in our Honors Analysis course (for undergraduate freshmen and sophomores), and all entering sophomores are continuing from the use of Spivak's Calculus and some baby Rudin that professors decide to implement into our Honors Calculus course. This is what leads me to believe that it's an easier transition into graduate analysis, which begins with big Rudin.

It should be fairly easy to see that big Rudin's approach is intended for very mature mathematicians, whereas Royden is intended for those attempting to make a transition from undergraduate analysis to graduate analysis. Royden starts out with defining not the general measure, but Lebesgue measure, and not until Part 3 does he begin general measure theory. In his preface to the second edition, Royden writes "The treatment of material given here is quite standard in graduate courses of this sort, although Lebesgue measure and Lebesgue integration are treated in this book before the general theory of measure and integration. I have found this a happy pedagogical practice, since the student first becomes familiar with an important concrete case and then sees that much of what he has learned can be applied in very general situations."

On the other hand, Rudin decides to introduce the theory of general measure rather early instead of concentrating on Lebesgue measure. I'd say that nearly all of the content of Royden's Real Analysis is covered in the first half of Big Rudin. So, in conclusion, I think both are very nicely written texts, both with very good problems, but they are intended for different readers. I'd say that the smartest route one would take would be: Spivak --> baby Rudin --> Royden --> big Rudin.

I have glanced at the contents of Royden, and the con's are that there's too much time spent on Topology, given that Baby Rudin already cover's a lot, such as some nontrivial facts like a metric space is compact if and only if every infinity subset has a limit point in it.

Well, most of the topology covered in Royden is quite different from baby Rudin. With chapter titles such as "Metric Spaces", "Topological Spaces", and "Compact Spaces", it may look elementary, but it's not. The chapter "Metric Spaces", for example, includes such topics as subspaces, Baire category, Absolute G-deltas, and the Ascoli-Arzela Theorem.

 

[morphism]

I've tried baby Rudin, big Rudin and his functional analysis book, and I've hated them all so much that I never finished much. (Althought big Rudin is by far the better of the three.) To me Rudin will always be a book where you can look up facts and find drill problems, and not a book from which to gain insight. Royden is better in this respect, and is a gentle transition from baby Rudin.

That said, it really depends on what you want to do next. If you want to learn some functional analysis, then I highly recommend Royden. He touches upon many of the fundamental ideas, such as locally convex spaces (including the Krein-Milman thm), the weak and weak-* topologies (including the Banach-Alaoglu thm), and many topics that Rudin doesn't mention. Topology is done well in Royden.

On the other hand, Rudin has a lot of things Royden doesn't, like material on Fourier analysis and complex analysis.

Personally I recommend this little gem: Topology and Modern Analysis by Simmons. It's divided into three parts: The first covers a lot of point-set topology, including things like Urysohn's lemma and metrization theorem, the Tietze extension theorem, Tychonoff, Stone-Weierstrass, Arzela-Ascoli, and the Stone-Cech compactification. The second part deals with the theory of Banach and Hilbert spaces, covering many of the basic facts of functional analysis, including finite dimensional spectral theory. The final part is a short survey of operator theory (with sections on some infinite dimensional spectral theory, C*-algebras, Gelfand-Neumark, Banach-Stone, etc.). It's really remarkably well-written. Although unfortunately it's not very extensive, and contains virtually no measure theory. But it goes very well with Royden! And both together provide a good bridge to Conway.

 

[rudinreader]

I don't think you are giving baby Rudin enough credit. For example, Baire's Theorem is a problem, one you would not want to skip, in baby Rudin... let's see if I can remember it in $$R^k$$? If $$R^k = \bigcup_k F_k$$, where $$F_k$$ is closed, then at least one has nonempty interior. BWOC, if they all have empty interior, fix e > 0 and consider $$N_e(0)$$. There must be a point in this ball that is not a limit point of $$F_1$$, so the closure of some $$N_{e_1}(x_1)$$ does not intersect $$F_1$$. Proceed inductively for $$F_k$$, and get $$\overline{N_{e_1}(x_1)} \supseteq ... \supseteq \overline{N_{e_k}(x_k)} \supseteq ...$$, where the infinite intersection is nonempty, but does not intersect $$\bigcup_{k} F_k = R^k$$, a contradiction.

Despite that only being a special case, along with the Ascoli Theorem for real functions in chapter seven, I'm just trying to point out that there is indeed a lot of topology covered in Baby Rudin, although a lot of students may skip some of the problems and thus miss out on it. A lot of the missing "general topology" detail can be supplemented in Chapter 1 of Bredon's Topology and Geometry, where in Bredon's words, "general topology is treated as much as most topology students would ever need, unless specializing in that area."

In Rudin's words in the preface to Adult Rudin, the first 7 chapters of baby Rudin is a sufficient prerequisite for Adult Rudin.

Sophomores study Royden at U of Chicago? Damn. What a head start!

My main point is that I think if you studied baby Rudin thouroughly, and if you realize that Adult Rudin is not really as hard as it looks (a lot of the problems in Chapter 1 just check that you have read the theorem statement, etc.), then Adult Rudin is the best.

But my opinion is still ****, because I haven't read either one as of yet.