# Principles of Mathematical Analysis by Walter Rudin

• Greg Bernhardt
In summary, "Principles of Mathematical Analysis" by Rudin is a concise and elegant undergraduate level textbook on real analysis. It is not suitable for first exposure to the subject, but rather serves as a reference for well-prepared readers. The book covers topics such as the real and complex number systems, basic topology, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, functions of several variables, integration of differential forms, and the Lebesgue theory. While the book has been praised for its elegance, it may be difficult to understand for some readers. There may also be a possible error in the proof of the least upper bound property and field axioms. Overall, "Principles of

## For those who have used this book

• Total voters
35

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[LIST]
[*] Preface[*] The Real and Complex Number Systems
[LIST]
[*] Introduction
[*] Ordered Sets
[*] Fields
[*] The Real Field
[*] The Extended Real Number System
[*] The Complex Field
[*] Euclidean Spaces
[*] Appendix
[*] Exercises
[/LIST][*] Basic Topology
[LIST]
[*] Finite, Countable, and Uncountable Sets
[*] Metric Spaces
[*] Compact Sets
[*] Perfect Sets
[*] Connected Sets
[*] Exercises
[/LIST][*] Numerical Sequences and Series
[LIST]
[*] 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
[/LIST][*] Continuity
[LIST]
[*] Limits of Functions
[*] Continuous Functions
[*] Continuity and Compactness
[*] Continuity and Connectedness
[*] Discontinuities
[*] Monotonic Functions
[*] Infinite Limits and Limits at Infinity
[*] Exercises
[/LIST][*] Differentiation
[LIST]
[*] 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
[/LIST][*] The Riemann-Stieltjes Integral
[LIST]
[*] Definition and Existence of the Integral
[*] Properties of the Integral
[*] Integration and Differentiation
[*] Integration of Vector-valued Functions
[*] Rectifiable Curves
[*] Exercises
[/LIST][*] Sequences and Series of Functions.
[LIST]
[*] 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
[/LIST][*] Some Special Functions
[LIST]
[*] Power Series
[*] The Exponential and Logarithmic Functions
[*] The Trigonometric Functions
[*] The Algebraic Completeness of the Complex Field
[*] Fourier Series
[*] The Gamma Function
[*] Exercises
[/LIST][*] Functions of Several Variables
[LIST]
[*] 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
[/LIST][*] Integration of Differential Forms
[LIST]
[*] Integration
[*] Primitive Mappings
[*] Partitions of Unity
[*] Change of Variables
[*] Differential Forms
[*] Simplexes and Chains
[*] Stokes' Theorem
[*] Closed Forms and Exact Forms
[*] Vector Analysis
[*] Exercises
[/LIST][*] The Lebesgue Theory
[LIST]
[*] 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
[/LIST][*] Bibliography[*] List of Special Symbols[*] Index
[/LIST]

• 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.

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(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?)

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!

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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.

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/SearchResults?an=serge+lang&sts=t&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.

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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.

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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.

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!

mathwonk said:
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.

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.

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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

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 Picard-Lindelof. 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.

astromme said:
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.

How do you circumnavigate around missing definitions? When the first (or second) chapter of a math book has missing definitions, what can one say?

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

@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.

Does he define what "open relative to" means?

Yes. Page 35 on the third edition, remark 2.29: "... to be quite explicit, let us say that $E$ is open relative to $Y$ if to each $p \in E$ there is associated an $r> 0$ such that $q \in E$ whenever $d(p,q) < r$ and $q \in Y$."

I would also point out that in context it's quite clear what open relative to means.

astromme said:
Yes. Page 35 on the third edition, remark 2.29: "... to be quite explicit, let us say that $E$ is open relative to $Y$ if to each $p \in E$ there is associated an $r> 0$ such that $q \in E$ whenever $d(p,q) < r$ and $q \in Y$."

I would also point out that in context it's quite clear what open relative to means.

Ok right, he does actually define it. But while the other definitions in the vicinity are given a definition block prefaced by "Definition:", that definition is buried in a block of text, the fourth sentence in a paragraph of six sentences. One could easily read the first few sentences and skip the rest because it is giving an example that one might already find trivial. Then seeing that the theorems on the next page use this "open relative to" idea, looking at the definition blocks doesn't help.

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.

@Verty I agree; the definition should probably have been lumped in with others. And I also agree that this book isn't the best for everyone; I just think it's working well for me. Note that I haven't learned/taught/referenced from other books, so I'm not sure my opinion should be given much credence. It certainly is possible that other analysis books would be even better for me than Rudin's - but at the moment things seem to be going well so I'm going to stick with it.

It's a good book, and it's meant to be read in a linear fashion. Don't complain about it if you skip parts of it. Oh, and for the prerequisites you don't need any of that nonsense. If you understand the rational number system, the binomial theorem, and some previous knowledge of trig functions will help, you should be good. You do not need to have taken any calc before. Rudin wisely rigorously develops it all for you.

I think the hardest part of the book is actually the second chapter, which is on metric spaces. I would do my best to avoid metric spaces and basic topology from that book for the first time, especially about the compact sets. As soon as the definition of a compact set is given (in terms of open covers), Rudin swiftly proceeds to myriads of lemmas without any motivation on why we even care about these particular sets. In the end, we reach the important Heine-Borel theorem, which gives us some concrete examples of compact sets on a euclidean space (on R^n, compact <=> closed and bounded), but the entire discussion on sequential compactness (which, in my opinion, is the analysts' notion of compactness) is hidden in the exercises.

Of course, everybody who reads the book will eventually learn about sequential compactness sooner or later, but I just find it strange how everything seemed "rushed" in that chapter. Was it supposed to be a review on metric space? Still brings me the bitter memory of learning analysis for the first time...

That being said, once I dealt with that chapter (as well as all the nitty-gritty about sequences and series in the next chapter), the rest of the book (at least up to chapter 8---never read anything beyond) was quite a treat. I thought it had good treatment of basic calculus on R^1, and I also enjoyed the chapter on the sequence of functions. It is a good book overall, but I found that particular chapter to be quite a pain to deal with (unless, of course, if one has seen metric space elsewhere!).

I've been through Chapters 1 and 2 and I like this book indeed. But, honestly, I don't think I would stand for it if I did not have the exposure to analysis I have acquired previously in the marvelous Bartle's ''Elements of Real Analysis''. So if Baby Rudin is your entry point in analysis and you don't feel comfortable with it, try Bartle.

Mathispuretruth said:
I've been through Chapters 1 and 2 and I like this book indeed. But, honestly, I don't think I would stand for it if I did not have the exposure to analysis I have acquired previously in the marvelous Bartle's ''Elements of Real Analysis''. So if Baby Rudin is your entry point in analysis and you don't feel comfortable with it, try Bartle.
This is actually a fantastic thing do do with really any subject; read multiple books, even side by side. It's good to see a subject treated a little differently, and for newer students, its good to see how an easier book can be "translated" into more rigorous and terse forms this way. Also even better is trying to prove each theorem on your own without looking at the book (to the best of your ability), or with minimal hints and peeps. It makes it so much more fun, and helped me get through hard concepts.

## 1. What is the main focus of "Principles of Mathematical Analysis" by Walter Rudin?

The main focus of "Principles of Mathematical Analysis" is to introduce students to the fundamental concepts of mathematical analysis, including topics such as real numbers, sequences and series, continuity, differentiation, and integration.

## 2. Is "Principles of Mathematical Analysis" suitable for beginners?

"Principles of Mathematical Analysis" is typically used as a textbook for upper-level undergraduate or graduate courses, so it may not be suitable for complete beginners. However, it does not assume any prior knowledge of analysis, so students with a strong foundation in calculus may find it accessible.

## 3. What sets "Principles of Mathematical Analysis" apart from other analysis textbooks?

One of the main features that sets "Principles of Mathematical Analysis" apart is its rigorous and concise approach to presenting mathematical concepts. The book also includes numerous challenging exercises to help students develop their problem-solving skills.

## 4. Can "Principles of Mathematical Analysis" be used as a reference book?

While "Principles of Mathematical Analysis" is primarily used as a textbook, it can also serve as a valuable reference for those with a strong background in analysis. Its clear and concise presentation of concepts makes it a useful resource for reviewing or refreshing one's understanding of key topics.

## 5. Are there any prerequisites for using "Principles of Mathematical Analysis"?

A strong foundation in calculus and mathematical proof techniques is recommended before using "Principles of Mathematical Analysis." Familiarity with basic concepts such as limits, continuity, and differentiation will also be helpful in understanding the material.

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