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# Principles of Mathematical Analysis by Walter Rudin

by Greg Bernhardt
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 Admin P: 8,429 Author: Rudin Title: Principles of Mathematical Analysis Amazon Link: http://www.amazon.com/Principles-Mat.../dp/007054235X Prerequisities: Rigorous calculus, including epsilon-delta proofs. Spivak's "Calculus" would be more than sufficient preparation. Table of Contents:  Preface The Real and Complex Number Systems Introduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix Exercises Basic Topology Finite, Countable, and Uncountable Sets Metric Spaces Compact Sets Perfect Sets Connected Sets Exercises Numerical Sequences and Series Convergent Sequences Subsequences Cauchy Sequences Upper and Lower Limits Some Special Sequences Series Series of Nonnegative Terms The Number e The Root and Ratio Tests Power Series Summation by Parts Absolute Convergence Addition and Multiplication of Series Rearrangements Exercises Continuity Limits of Functions Continuous Functions Continuity and Compactness Continuity and Connectedness Discontinuities Monotonic Functions Infinite Limits and Limits at Infinity Exercises Differentiation The Derivative of a Real Function Mean Value Theorems The Continuity of Derivatives L'Hospital's Rule Derivatives of Higher Order Taylor's Theorem Differentiation of Vector-valued Functions Exercises The Riemann-Stieltjes Integral Definition and Existence of the Integral Properties of the Integral Integration and Differentiation Integration of Vector-valued Functions Rectifiable Curves Exercises Sequences and Series of Functions. Discussion of Main Problem Uniform Convergence Uniform Convergence and Continuity Uniform Convergence and Integration Uniform Convergence and Differentiation Equicontinuous Families of Functions The Stone-Weierstrass Theorem Exercises Some Special Functions Power Series The Exponential and Logarithmic Functions The Trigonometric Functions The Algebraic Completeness of the Complex Field Fourier Series The Gamma Function Exercises Functions of Several Variables Linear Transformations Differentiation The Contraction Principle The Inverse Function Theorem The Implicit Function Theorem The Rank Theorem Determinants Derivatives of Higher Order Differentiation of Integrals Exercises Integration of Differential Forms Integration Primitive Mappings Partitions of Unity Change of Variables Differential Forms Simplexes and Chains Stokes' Theorem Closed Forms and Exact Forms Vector Analysis Exercises The Lebesgue Theory Set Functions Construction of the Lebesgue Measure Measure Spaces Measurable Functions Simple Functions Integration Comparison with the Riemann Integral Integration of Complex Functions Functions of Class $\mathcal{L}^2$ Exercises Bibliography List of Special Symbols Index  User comments: jbunniii For the well prepared reader, this is a beautifully clear treatment of the main topics of undergraduate real analysis. Yes, it is terse. Yes, the proofs are often slick and require the reader to fill in some nontrivial gaps. No, it doesn't spend much time motivating the concepts. It is not the best book for a first exposure to real analysis - that honor belongs to Spivak's "Calculus." But don't kid yourself that you have really mastered undergraduate analysis if you can't read Rudin and appreciate its elegance. It also serves as a nice, clean, uncluttered reference which few graduate students would regret having on their shelves. micromass This is a wonderful book iff you can handle it. Do not use Rudin as your first exposure to analysis, it will be a horrible experience. However, if you already completed a Spivak level text, then Rudin will be a wonderful experience. It contains many gems and many challenging problems. Personally, I find his approach to differential forms and Lebesgue integration quite weird though. I think there are many books that cover it better than him. But the rest of the book is extremely elegant and nice.
 P: 73 (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 $q \in \alpha$, then $-q \notin \beta$. Therefore $\beta \neq \mathbb{Q}$" but this is not true. For example, say our cut $\alpha$ was bounded above by -2 (not inclusive). Then, $-3$ would be in $\alpha$, but $3$ would be in $\beta$. Instead, the following reasoning is valid: Choose $p \in \mathbb{Q}$ such that $-p \in \alpha$. Then, since for any $r > 0$, $-p-r < -p$, and so $-p-r \in \alpha$ by (II). Since $\beta$ consists of all those $p$ such that $\exists \, \beta \in \mathbb{R}$ such that $-p-r \notin \alpha$, we conclude $p \notin \beta$. Well, now I'm confused because I feel as though my counterexample is correct, but I've just proven that if $-p \in \alpha$, then $p \notin \beta$, which is clearly the same as $p \in \alpha$, then $-p \notin \beta$. (Or is it?)
 Sci Advisor HW Helper P: 9,406 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!
P: 73

## 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.
 Sci Advisor HW Helper P: 9,406 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.
 Sci Advisor HW Helper P: 9,406 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.
 P: 73 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.
 Sci Advisor HW Helper P: 9,406 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!
P: 376
 Quote by mathwonk 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.
Read Rudin's treatment of L'Hôpital's/Bernoulli rule and you will swiftly discover that his main intention behind this book was not pedagogical.
 P: 46 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 epsilon-delta 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 non-trivial 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.
 P: 639 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

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