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Analysis Principles of Mathematical Analysis by Walter Rudinby Greg Bernhardt
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#1
Jan1813, 10:17 PM

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P: 9,698

Table of Contents:



#2
Feb1013, 12:07 PM

P: 74

(If this is not the right place for this discussion, please excuse my post, and I will happily have the mods move it to the right place.)
Possible erratum: On p. 19 of the 3rd edition, in the proof that the real numbers have the least upper bound property and the field axioms, Rudin states: "If [itex]q \in \alpha[/itex], then [itex]q \notin \beta[/itex]. Therefore [itex]\beta \neq \mathbb{Q}[/itex]" but this is not true. For example, say our cut [itex]\alpha[/itex] was bounded above by 2 (not inclusive). Then, [itex]3[/itex] would be in [itex]\alpha[/itex], but [itex]3[/itex] would be in [itex]\beta[/itex]. Instead, the following reasoning is valid: Choose [itex]p \in \mathbb{Q} [/itex] such that [itex]p \in \alpha [/itex]. Then, since for any [itex] r > 0 [/itex], [itex] pr < p [/itex], and so [itex] pr \in \alpha [/itex] by (II). Since [itex]\beta [/itex] consists of all those [itex]p[/itex] such that [itex]\exists \, \beta \in \mathbb{R} [/itex] such that [itex] pr \notin \alpha [/itex], we conclude [itex]p \notin \beta [/itex]. Well, now I'm confused because I feel as though my counterexample is correct, but I've just proven that if [itex] p \in \alpha [/itex], then [itex] p \notin \beta [/itex], which is clearly the same as [itex] p \in \alpha [/itex], then [itex] p \notin \beta [/itex]. (Or is it?) 


#3
Feb1213, 01:12 PM

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I think most people know this is not my favorite math book. I taught out of it one year in the senior math major course and the students suffered tying to understand it. As for myself, in every math book I like there is some topic that I have learned and that I remember that book for.
E.g. from Courant I have always remembered the explanation of the correspondence between real numbers and points on a line, and the clear criteria for series convergence, and the footnote where formula for the sum of the first n kth powers is derived, and the principle of the point of accumulation,..... From Van der Waerden I recall a succinct account of the definition of the real numbers, as a quotient ring of the ring of Cauchy sequences of rationals, modded out by the maximal ideal of null sequences. From Lang Analysis I I remember the crystal clear account of the Riemann integral as characterized simply by being monotone and additive over intervals and the area of a rectangle as base times height,.. By contrast, there is virtually no topic that I can recall learning from Rudin's book, and some topics like differential forms mentioned by micromass, simply should not be learned here. Well maybe I learned a little something about Dedekind cuts. But to me most explanations here are simply not memorable. He succeeds in impressing me how smart and clever he is, but not in instructing me. To try to be somewhat fair, it is entirely possible that my problem with Rudin is that I am a geometer and he is an analyst (his discussion of differential forms takes out all the geometry), and analysis has always seemed to me like a book of seven seals, but i think think that is partly because of books like this one. On the other hand if you like this book, take heart, maybe you are a future analyst! 


#4
Feb1213, 02:07 PM

P: 74

Principles of Mathematical Analysis by Walter Rudin
Thank you for your reply, mathwonk. Do you have any analysis book in particular you would recommend? My analysis class is using the book by Protter and Morrey, which has a strong first half but is weak on multivariable. I was thinking about using Stricharz? I also have the Dover books by Rosenlicht and Komolgorov.



#5
Feb1213, 02:20 PM

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Well, if you like and can learn from Rudin, then it may be for you, but it is not my cup of tea. I probably should hesitate more to recommend against it for others, because I think every analysis prof I asked has liked Rudin. So for analysis minded folks this may be it.
I'm not sure what part of analysis you ask about, but presumably the same parts that are in Rudin, i.e. rigorous calculus. I like the books by Wendell Fleming, and Michael Spivak, as well as maybe Williamson, Crowell and Trotter for calculus. Fleming even does a nice job on Lebesgue integration, and Spivak does a fine job on differential forms. WC&T is a several variable calc book. And I have liked almost every book by Sterling K Berberian, as he is a superb expositor who tries actually to teach the reader. I also liked Lang's Analysis I, now called something else, like Undergraduate Analysis (and more expensive). http://www.abebooks.com/servlet/Sear...&tn=analysis+I Rosenlicht and Kolmogorov are also famously excellent expositors. As always, go to the library and peruse until one strikes you as clear. 


#6
Feb1213, 02:33 PM

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here is an example of what i dislike in Rudin compared to say Courant. On page 65, Rudin begins the ratio and root tests with a brutal statement of a theorem, no motivation at all, no explanation. What is he doing? You have to study the proof to find out.
Courant on page 377 explains that "All such considerations of convergence depend on comparison of the series in question with a second series,...,chosen in such a way that its convergence can be readily tested." Then on the next page he begins with the simplest case, comparison with a geometric series. Two slightly different methods of making the comparison lead to the root and ratio tests, which are thus seen clearly as special cases of a simple general principle. Rudin's arguments are essentially the same but there is no helpful discussion in words to make it memorable. I.e. Rudin plunges into the details with no preliminary statement of what the idea is behind the argument to come. For me, once I understand the idea I can provide the details myself. It is harder to go backwards from the details to reconstruct what the idea was, although indeed it is there hidden under the argument. I would compare it to a magician explaining a trick by saying: first my beautiful assistant comes out and distracts your attention by displaying something while I reach under my arm for a compressed bouquet of flowers which I then expand before your eyes. Or else he just performs the trick and you are left to wonder what happened. But everyone has a different learning style. Some people like Rudin very much. 


#7
Feb1213, 03:05 PM

P: 74

Thank you for being so thorough. I think I will try to use some combination of Lang, Apostol, Spivak and Courant, and then move on to Rudin and eventually Berberian.



#8
Feb1213, 08:35 PM

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remember what i actually said: don't take my advice on exactly which book to use, take my advice to look at them yourself, and then make your own choice. give yourself some credit, you can make an informed decision, if you inform yourself first. enjoy!



#9
Apr713, 11:35 PM

P: 376




#10
Apr713, 11:48 PM

P: 47

I didn't have any exposure to analysis when I started reading Rudin, but it was at least somewhat readable. I don't think it's necessary to take a calculus course with epsilondelta proofs before taking (though that would be very helpful), but having some experience with proofs was important. When I first read it, I found it to be very dense and I often had to draw pictures to have any idea of what is going on. On the second pass, I found it was very concise and contained a lot of elegant proofs. Also, the problems were a lot of fun to work on. However, I was still frustrated that a lot of steps that were nontrivial to me were omitted as "obvious". As for the content itself, I think most of the criticism relates to the last couple of chapters. I'm not really qualified to comment on this as I haven't read them, but I think most places don't use (this) Rudin for courses relating to the content of these chapters and only use the book for an intoductory analysis course.



#11
Sep1613, 08:17 PM

P: 698

Here is a perspective of an electrical engineer who wanted to teach himself analysis a handful of years ago, mostly for fun. I don''t think my background is too unusual for an engineer that was not in an "applied math" branch of EE (many of whom I know took an analysis course based on Rudin). The closest thing I ever took to a "real" math course was a senior level applied complex analysis course from the math department, where I did have to do some proofs as part of each homework assignment, and did indeed appreciate the challenge. I first picked up Rudin because my wife has a BS in math so the book was already at home. Least resistance and all that. Anyway, it didn't take long before I gave up. I really did need more to help me on my way, and it can get frustrating to not understand the motivation or general approach of a proof before he simply reveals a proof.
I ended up learning elementary analysis from "analysis and an introduction to proof" by Lay, supplemented by "mathematical analysis: a straightforward approach" by Binmore and the first 7 chapters of "the way of analysis" by Strichartz. After that I could pick up Rudin and appreciate it much more, and can recognize how hard some of the problems are. So I cringe when I see folks suggest Rudin for people first learning analysis on their own  without someone to help (or adequate preparation) many of us are not smart enough to handle it and will get discouraged. I do not know what the best books are for folks like me, I just know that the books I used were much preferred to Rudin. If I get the urge again sometime I may give it another try with my improved background, but then again maybe not. Strichartz seems much easier for me to read to understand the what/why/how of the subject, even if he does meander a bit. jason 


#12
Jul3114, 02:18 AM

P: 5

It might be helpful for someone somewhere to know that I didn't complete a book like Spivak before starting Rudin and it's going fine. On the other hand, I had built up math maturity from linear algebra and our differential equations course which, at a high point of abstraction proved PicardLindelof. Also I had seen delta epsilon proofs throughout the course, but we didn't have to do too many of them ourselves. I'm not sure how different this sort of preparation is compared to what one might have from Spivak, but when I started reading Spivak after I had finished this course I found most of the problems too straightforward and thus decided to skip ahead to Rudin.
About Rudin: I can honestly say this is the best book I've ever read and I'm only 80 pages in. His pace is relentless and the lack of details in his proofs is almost riduculous at points, but this is completely circumnavigated by trying (and hopefully succeeding most of the time) to prove the Theorems yourself. In this fashion you will build up a strong idea of what the underlying structure is, even if you can't prove it yourself, and thus will be much better prepared to understand what he is leaving out. I would also point out that one of my favorite consequences of Rudin's notorious brevity is that one feels that one is making an enormous amount of progress by completing just 20 pages. Wheras with a book like Apostol's Calculus there are two essentially infinite volumes where completing 20 pages is a drop in the bucket! In short, it is much easier to stay motivated when reading Rudin. Oh yeah, and the problems are very very good. Try to solve as many as you have time for. Overall impression: This book is not for teaching analysis. This book is for teaching you how to think. 


#13
Jul3114, 02:20 PM

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#14
Jul3114, 02:36 PM

P: 47

What do you consider to be missing definitions in Rudin? I think the comment above was specifically about missing steps in proofs.



#15
Jul3114, 04:11 PM

P: 5

@verty I was speaking specifically about missing steps in proofs. On the contrary with definitions, I have only thought twice that he didn't define something; both times when I looked back it turns out that the mistake was with me as I didn't read closely enough.



#16
Aug114, 12:49 AM

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Does he define what "open relative to" means?



#17
Aug114, 12:56 AM

P: 5

Yes. Page 35 on the third edition, remark 2.29: "... to be quite explicit, let us say that [itex] E [/itex] is open relative to [itex]Y[/itex] if to each [itex] p \in E[/itex] there is associated an [itex] r> 0 [/itex] such that [itex] q \in E [/itex] whenever [itex] d(p,q) < r [/itex] and [itex] q \in Y [/itex]."
I would also point out that in context it's quite clear what open relative to means. 


#18
Aug114, 01:24 AM

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On page 32, ten definitions are given in one block, including "open" and "closed". It certainly gives one the impression that all the relevant definitions will be given together in one block. Then early on page 35, there is a definition block with one definition. So one might certainly conclude that no further definitions appear on that page. Rudin, if he tried to hide that definition, could hardly do a better job. Obviously this is a matter of opinion, because to you (Astromme) it was obvious in context. Each to his own, I suppose. 


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