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Principles of Mathematical Analysis
Principles of Mathematical Analysis Second Edition was written by Walter Rudin, published by McGraw-Hill Book Company and was printed in 1964 in a Hardcover binding.
270 pages, Hardcover
First published January 1, 1964
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Displaying 1 - 30 of 106 reviews
November 8, 2015
A remarkable text, but on reflection, perhaps not the most helpful read by itself- unless frustration is just your kind of motivator.
Here are some supplementary resources-
(lectures based on this text)
(companions for the text)
The following can also be used to supplement Rudin or as alternatives:
by Amann and Escher (first in a three volume sequence- the more I refer to these the more impressed I am)
by Tao
(first in a two volume set- also see his and-
)
Abbott, Rosenlicht, Pugh, Kolmogorov, Dieudonne, Hardy, Landau, Shilov, Courant, Apostol, etc. could also be profitably considered in this capacity.
After this? A natural course is to move on to Rudin's own (complemented nicely by the 4th edition of Royden; Bressoud's other book might also be of interest for its historically informed narrative; to gain more motivation for topology try Janich's text or something like ). Then one would graduate to a course in functional analysis (again, there is Rudin's text, but I don't like it for that as much as this and Real and Complex Analysis, rather try Haim Brezis).
Here are some advanced resources-
These as well as other classic mathematics texts may not be all that reasonably priced, in which case you might consider having them ordered from South Asia or China where they sell for a special rate. I'm uncertain whether this is allowed or if dealers in the region will venture it even if a loophole existed, and delivery costs may be heavy, but the purchase ought to be cheaper even so considering the obscene asking prices for some of these textbooks.
Here are some supplementary resources-
(lectures based on this text)
(companions for the text)
The following can also be used to supplement Rudin or as alternatives:
by Amann and Escher (first in a three volume sequence- the more I refer to these the more impressed I am)
by Tao
(first in a two volume set- also see his and-
)
Abbott, Rosenlicht, Pugh, Kolmogorov, Dieudonne, Hardy, Landau, Shilov, Courant, Apostol, etc. could also be profitably considered in this capacity.
After this? A natural course is to move on to Rudin's own (complemented nicely by the 4th edition of Royden; Bressoud's other book might also be of interest for its historically informed narrative; to gain more motivation for topology try Janich's text or something like ). Then one would graduate to a course in functional analysis (again, there is Rudin's text, but I don't like it for that as much as this and Real and Complex Analysis, rather try Haim Brezis).
Here are some advanced resources-
These as well as other classic mathematics texts may not be all that reasonably priced, in which case you might consider having them ordered from South Asia or China where they sell for a special rate. I'm uncertain whether this is allowed or if dealers in the region will venture it even if a loophole existed, and delivery costs may be heavy, but the purchase ought to be cheaper even so considering the obscene asking prices for some of these textbooks.
March 14, 2012
It's the classic. Terse, direct, clear, and horribly painful.
This book forms the basis for the first class in real analysis (in a single variable) for countless thousands of hapless students who decide to concentrate on math. It's chosen by professors who have had decades of experience as university mathematicians, and have achieved a certain Zen-like understanding of the knowledge contained within.
It's too bad they forget what it was like to receive that knowledge.
As the purchaser and consumer of this text, you don't really have a choice. This is the gateway that those in knowledge and power have chosen for your entre to this field.
There are those that will love this book, find themselves within it, and savor every moment they spend turning the pages to new realms of knowledge.
To the rest of us, I recommending finding one of those people and kidnapping them. Force them to do things they are unfamiliar with, things that are illogical and unwholesome. Then ask them to tutor you.
This book forms the basis for the first class in real analysis (in a single variable) for countless thousands of hapless students who decide to concentrate on math. It's chosen by professors who have had decades of experience as university mathematicians, and have achieved a certain Zen-like understanding of the knowledge contained within.
It's too bad they forget what it was like to receive that knowledge.
As the purchaser and consumer of this text, you don't really have a choice. This is the gateway that those in knowledge and power have chosen for your entre to this field.
There are those that will love this book, find themselves within it, and savor every moment they spend turning the pages to new realms of knowledge.
To the rest of us, I recommending finding one of those people and kidnapping them. Force them to do things they are unfamiliar with, things that are illogical and unwholesome. Then ask them to tutor you.
December 17, 2009
'Baby Rudin' is to learning analysis as artificial insemination is to sexual reproduction. It's not good, it's not bad, and there are more fun ways of going about it, but at least it gets the job done.
October 9, 2009
my god. this is the best mathematics book i've ever had a pleasure of using, even if it totally reduced my brain to oatmeal. don't let the size of the book deceive you (it's much smaller than most math textbooks.) it's incredibly terse, the problem sets are "fucking bullshit" (mind you, this is what me and my classmates would usually say after trying to attack a single problem for about a good couple of hours - but as "fucking bullshit" the problems were, they are pedagogically EXCELLENT problems with no problem sets seeming to be trite and useless), the book is quite self-contained and can be approached with enough mathematical maturity in calculus and perhaps a little bit of advanced calculus, etc.
EVERYONE should read this book and trudge through it properly. by properly, i mean skimming through each section, and then going through each proposition and theorem very slowly and proving each one on your own without looking at how rudin proves it.
EVERYONE should read this book and trudge through it properly. by properly, i mean skimming through each section, and then going through each proposition and theorem very slowly and proving each one on your own without looking at how rudin proves it.
January 23, 2022
For the honors student and aspiring mathematician, freshman year in college represents a right of passage, at least at the top-tier universities. He must set aside slovenly habits picked up in high school and learn to wield the predicate calculus in all its glory in order to construct rigorous proofs of non-trivial theorems. For many, this can be an excruciating sink-or-swim exercise. Analysis, on the foundation on which it was put by Cauchy and Weierstrass in the nineteenth century, involves arcane epsilontics that do not come naturally, although once acquired they constitute an effective tool in the analyst’s arsenal. For instance, a function f(x) is continuous at x if for all ε>0, there exists a δ>0 such that |f(x)-f(y)|<ε whenever |x-y|<δ. Statements of this kind involving concatenation of universal and existential quantifiers are common, and this one is among the simplest (merely to state the property of Cauchy completeness, for instance, involves a fearsome-looking proposition containing no less than eight interdependent quantifiers, five universal and three existential, plus one logical implication: ∀∃∀∀⇒∃∀∃∀). To the neophyte unaccustomed to such manipulations, an operation such as negating a statement containing more than one quantifier can be hard to perform, but practice is needed to become proficient at proving typical results (for instance, one will gather experience as to what direction to take in showing an implication, either directly or in the contrapositive). To acquire facility at juggling statements involving quantifiers and logical implication calls for a leap of the imagination, from monadic logic (Aristotle to Kant) to polyadic logic (post-Fregean).
At most schools, the sorting out of the wheat from the chaff takes place during an accelerated course in introductory analysis, which is well suited to the purpose (in recent years, at some places a preparatory course on techniques of proof such as induction has come to be inserted beforehand, but in this reviewer’s opinion that is just a waste of good students� time). This reviewer’s textbook for the spring semester of his freshman year was Walter Rudin’s legendary Principles of Mathematical Analysis. Affectionately known as ‘baby Rudin� in order to distinguish it from the graduate-level textbook on real and complex analysis by the same author (‘papa Rudin�), this text has been a staple in the curriculum for generations and many will swear by it. It takes its origin from when, as a graduate student back in the 1950’s, Rudin had to teach analysis to advanced undergraduates and, finding there to be no satisfactory text, decided to write his own. Over the course of his career he tinkered with and perfected it, and it has achieved classic status in its third and final edition (in the McGraw-Hill international series in pure and applied mathematics, the entries of which are outrageously expensive, by the way).
Sketch of contents: the first chapter is given over to Dedekind’s arithmetical construction of the real number line from the rationals and the least-upper-bound property (it can easily be skipped). The course gets underway in earnest in chapter two with basic topology in metric spaces and properties of denumerable sets. Chapter three covers numerical sequences and series (convergence, root and ratio tests, rearrangements). These simple notions, later to be generalized to other spaces than just the real number line, are basic to all of analysis. At last in chapter four we arrive at the epsilon-delta definition of limits, continuity, compactness, uniform continuity, connectedness and infinity. Here is where the going gets hard for the inexperienced student. Chapters five and six introduce the derivative and the Riemann-Stieltjes integral. The material will be for the most part familiar to those who have had calculus at the high-school level (at least formally), but the proofs entirely novel. The same cannot be said of the next two chapters, seven and eight, on sequences and series of functions. The old and trusted concepts are now applied to function spaces equipped with a metric. Any serious mathematician will have to master this abstract point of view sooner or later (it dates to the early decades of the twentieth century, a revolutionary period when the foundations of topology and functional analysis in their modern form were laid). As an index of the complications one encounters, one meets the need for the concepts of uniform convergence and how it relates to continuity, differentiation, equicontinuous families and compactness. Chapter seven culminates in a nice and thorough proof of the Stone-Weierstrass theorem on approximation of continuous functions by polynomials, a foundational result of analysis. Chapter eight covers special functions such as the exponential, logarithm, gamma function and elementary Fourier series.
Everyone is agreed that the first eight chapters in Rudin’s textbook constitute almost the canonical exposition of introductory analysis. There seems to be a consensus that the remaining three chapters on multivariate functions, integration of differential forms and Stokes� theorem and beginning measure theory up to the Lebesgue integral are less strong. Certainly, there are other good places to get acquainted with these subjects, but in this reviewer’s opinion, however, there is nothing wrong with Rudin’s treatment, although to be sure it looks more like an analyst’s view than a geometer’s. The big theorems receive efficient proofs: implicit function theorem, inverse function theorem, rank theorem, Stokes� theorem. As to the latter, it should be remarked that Rudin employs an idiosyncratic definition of differential forms that is just sufficient to enable him to obtain their elementary properties and derive Stokes� theorem, but which completely elides their intuitive geometrical significance, which one would learn in a standard course on differentiable manifolds (no manifolds here, everything is done in an open subset of Euclidean space). Suffice it to say that these three chapters are too concise and by no means adequate to their respective subjects. This observation is particularly true of the last chapter, which breezes through measure theory and the Lebesgue integral up to the Riesz-Fischer theorem in just thirty pages! Half of an ordinary textbook on real analysis could be devoted to these subjects alone. If one is willing to bear with Rudin and view these chapters on their own terms as a thought-experiment, possibly their value, which consists in seeing how Rudin does things, could be appreciated. The problems in these chapters are just as good as in the earlier ones.
What sets ‘baby Rudin� apart from the crowd is its excellent style. Far more than the physicist, the mathematician is apt to set to work as a craftsman and to hold himself responsible to produce a work of art, and Rudin is an acknowledged master. In a scientific field in which knowledge is to be conveyed, the medium cannot be the entire message, but it is certainly a part of it. The hallmarks of Rudin’s style are concision and elegance; he will not content himself with proving a result in any which way, but strives for a pleasing economy. In contrast, Pugh and to a lesser extent Apostol and Abbott in their textbooks on introductory analysis seem to take the view that the more said, the more help it will be to the student (especially the beginner or less proficient). Now, in any technical exposition, there will always be considerable leeway for a judgment call about what to leave tacit; authors differ, but when everything is thrown in in the hope that enough if it will stick in the reader’s mind and facilitate his comprehension, the effect can be rather tedious (Arnold, on the other hand, is an example of someone who says too little for ordinary mortals, leaving them with the illusion of understanding without adequate erudition). Rudin takes care to put in enough, but not too much. The discerning reader should be on the lookout for subtleties and when one is encountered, trust that he can resolve it to his own satisfaction with a moment’s thought. The budding mathematician must ever cultivate his sense of mathematical maturity and here is a suitable place to begin, at the level of introductory analysis. Such attention to detail is really part of the art of any communication, as Fowler tirelessly seeks to get across throughout his Modern English Usage.
The homework exercises—some 289 of them over eleven chapters—are legendary for their difficulty, but this may be mainly a function of the fact that callow students are surprised when called upon to prove something for real for the first time. In retrospect to someone who has gained exposure to the subject at the graduate level, they appear often rather challenging but not quite impossible if one be willing to make a determined effort. This reviewer remembers, for instance, wrapping his mind for hours around the problem to find a non-empty perfect set on the real line that contains no rational number. Rudin’s problems tend to be conceptual rather than computational; they are selected to illustrate a point and supporting calculations are secondary in importance and only very seldom do they become very involved. In contrast, many problems in Courant and John seem to have little justification other than to prove that one can see a demanding and byzantine calculation through to its successful conclusion, as if mastering mathematics were a signaling game. For a nice specimen of Rudin’s preferences, see Problem 7.14 in which one is asked to show that a given continuous map takes the real line onto the unit square (in fact, it maps the Cantor set onto the unit square).
In sum: want to become a real mathematician? Hit the books hard with the classic baby Rudin or condemn yourself forever to mediocrity!
At most schools, the sorting out of the wheat from the chaff takes place during an accelerated course in introductory analysis, which is well suited to the purpose (in recent years, at some places a preparatory course on techniques of proof such as induction has come to be inserted beforehand, but in this reviewer’s opinion that is just a waste of good students� time). This reviewer’s textbook for the spring semester of his freshman year was Walter Rudin’s legendary Principles of Mathematical Analysis. Affectionately known as ‘baby Rudin� in order to distinguish it from the graduate-level textbook on real and complex analysis by the same author (‘papa Rudin�), this text has been a staple in the curriculum for generations and many will swear by it. It takes its origin from when, as a graduate student back in the 1950’s, Rudin had to teach analysis to advanced undergraduates and, finding there to be no satisfactory text, decided to write his own. Over the course of his career he tinkered with and perfected it, and it has achieved classic status in its third and final edition (in the McGraw-Hill international series in pure and applied mathematics, the entries of which are outrageously expensive, by the way).
Sketch of contents: the first chapter is given over to Dedekind’s arithmetical construction of the real number line from the rationals and the least-upper-bound property (it can easily be skipped). The course gets underway in earnest in chapter two with basic topology in metric spaces and properties of denumerable sets. Chapter three covers numerical sequences and series (convergence, root and ratio tests, rearrangements). These simple notions, later to be generalized to other spaces than just the real number line, are basic to all of analysis. At last in chapter four we arrive at the epsilon-delta definition of limits, continuity, compactness, uniform continuity, connectedness and infinity. Here is where the going gets hard for the inexperienced student. Chapters five and six introduce the derivative and the Riemann-Stieltjes integral. The material will be for the most part familiar to those who have had calculus at the high-school level (at least formally), but the proofs entirely novel. The same cannot be said of the next two chapters, seven and eight, on sequences and series of functions. The old and trusted concepts are now applied to function spaces equipped with a metric. Any serious mathematician will have to master this abstract point of view sooner or later (it dates to the early decades of the twentieth century, a revolutionary period when the foundations of topology and functional analysis in their modern form were laid). As an index of the complications one encounters, one meets the need for the concepts of uniform convergence and how it relates to continuity, differentiation, equicontinuous families and compactness. Chapter seven culminates in a nice and thorough proof of the Stone-Weierstrass theorem on approximation of continuous functions by polynomials, a foundational result of analysis. Chapter eight covers special functions such as the exponential, logarithm, gamma function and elementary Fourier series.
Everyone is agreed that the first eight chapters in Rudin’s textbook constitute almost the canonical exposition of introductory analysis. There seems to be a consensus that the remaining three chapters on multivariate functions, integration of differential forms and Stokes� theorem and beginning measure theory up to the Lebesgue integral are less strong. Certainly, there are other good places to get acquainted with these subjects, but in this reviewer’s opinion, however, there is nothing wrong with Rudin’s treatment, although to be sure it looks more like an analyst’s view than a geometer’s. The big theorems receive efficient proofs: implicit function theorem, inverse function theorem, rank theorem, Stokes� theorem. As to the latter, it should be remarked that Rudin employs an idiosyncratic definition of differential forms that is just sufficient to enable him to obtain their elementary properties and derive Stokes� theorem, but which completely elides their intuitive geometrical significance, which one would learn in a standard course on differentiable manifolds (no manifolds here, everything is done in an open subset of Euclidean space). Suffice it to say that these three chapters are too concise and by no means adequate to their respective subjects. This observation is particularly true of the last chapter, which breezes through measure theory and the Lebesgue integral up to the Riesz-Fischer theorem in just thirty pages! Half of an ordinary textbook on real analysis could be devoted to these subjects alone. If one is willing to bear with Rudin and view these chapters on their own terms as a thought-experiment, possibly their value, which consists in seeing how Rudin does things, could be appreciated. The problems in these chapters are just as good as in the earlier ones.
What sets ‘baby Rudin� apart from the crowd is its excellent style. Far more than the physicist, the mathematician is apt to set to work as a craftsman and to hold himself responsible to produce a work of art, and Rudin is an acknowledged master. In a scientific field in which knowledge is to be conveyed, the medium cannot be the entire message, but it is certainly a part of it. The hallmarks of Rudin’s style are concision and elegance; he will not content himself with proving a result in any which way, but strives for a pleasing economy. In contrast, Pugh and to a lesser extent Apostol and Abbott in their textbooks on introductory analysis seem to take the view that the more said, the more help it will be to the student (especially the beginner or less proficient). Now, in any technical exposition, there will always be considerable leeway for a judgment call about what to leave tacit; authors differ, but when everything is thrown in in the hope that enough if it will stick in the reader’s mind and facilitate his comprehension, the effect can be rather tedious (Arnold, on the other hand, is an example of someone who says too little for ordinary mortals, leaving them with the illusion of understanding without adequate erudition). Rudin takes care to put in enough, but not too much. The discerning reader should be on the lookout for subtleties and when one is encountered, trust that he can resolve it to his own satisfaction with a moment’s thought. The budding mathematician must ever cultivate his sense of mathematical maturity and here is a suitable place to begin, at the level of introductory analysis. Such attention to detail is really part of the art of any communication, as Fowler tirelessly seeks to get across throughout his Modern English Usage.
The homework exercises—some 289 of them over eleven chapters—are legendary for their difficulty, but this may be mainly a function of the fact that callow students are surprised when called upon to prove something for real for the first time. In retrospect to someone who has gained exposure to the subject at the graduate level, they appear often rather challenging but not quite impossible if one be willing to make a determined effort. This reviewer remembers, for instance, wrapping his mind for hours around the problem to find a non-empty perfect set on the real line that contains no rational number. Rudin’s problems tend to be conceptual rather than computational; they are selected to illustrate a point and supporting calculations are secondary in importance and only very seldom do they become very involved. In contrast, many problems in Courant and John seem to have little justification other than to prove that one can see a demanding and byzantine calculation through to its successful conclusion, as if mastering mathematics were a signaling game. For a nice specimen of Rudin’s preferences, see Problem 7.14 in which one is asked to show that a given continuous map takes the real line onto the unit square (in fact, it maps the Cantor set onto the unit square).
In sum: want to become a real mathematician? Hit the books hard with the classic baby Rudin or condemn yourself forever to mediocrity!
March 20, 2008
This was the text book for my real analysis course. I did not like this text book. I found it dense and hard to gain relevant information from it. Rudin optimized this book for the shortest proofs, and that isn't particularly useful for learning the material. Important results in Mathematics are given as exercises, which is cool, except that they are really hard, and there isn't enough material in the book to cover the problems.
December 26, 2018
This is a mesmerizing text that develops real and complex numbers, topology, algebra, and calculus first in one, and then in multiple variables. I'm not a math major and I skipped most of the exercises, but the ones I did were worth the effort. Baby Rudin elegantly presents, proves, and discusses all its material and, without waxing poetic, it shows you how beautiful math can be.
August 19, 2013
The title of this calculus bible should have been "So you are a mathematical genius".
Rudin introduces everything as if they came from out of nowhere, like a black hole. Counter-intuitive! Too bad!
Rudin introduces everything as if they came from out of nowhere, like a black hole. Counter-intuitive! Too bad!
June 30, 2020
Very clear. Although it's terse, I found it much easier to read than Dummit and Foote. Very good excersices(although I must confess I've done very little). That said, some methods of proof rely on trickery and are unmotivated.
April 2, 2019
most treasured memory in years studying math
May 3, 2023
So informative it almost killed me.
June 17, 2024
I've had moment the past year where this book has been frustrating to read and moments where it's been an amazing resource. On one hand, some of the proofs have very short explanations that make them hard to follow, but at the same time, trying to decode how they work might just be part of the education this book provides.
May 30, 2017
Analysis, the hard way. Definitely worth the effort, though.
Read
July 18, 2015I have mixed feelings about this book. How to describe it... ok, let's talk kung-fu movies. So there's a standard trope in martial arts movies where the young apprentice shows up at the stoop of the Old Master and says, "teach me to fight". And the Old Master decides that instead of doing the obvious thing and having our hapless padawan practice something reasonable like, you know, punching techniques, the Old Master tells the aspirant do a series of incomprehensible and difficult tasks. Carrying the Old Master up and down the mountains. Knitting sweaters while hanging upside-down over hot coals. Doing the Old Master's laundry. And so on. Usually, it's never clear if the training is difficult because Sensei is trying to impart some kind of deeper wisdom or if he's really just a resentful old jerk who takes pleasure in making young students suffer.
Principles of Mathematical Analysis is the Old Master. It is completely uncompromising � no diagrams, the proofs are often opaque, the definitions unmotivated � and the book carries more than a whiff of that sadistic strain in math education that sees formal rigor and a lack of justification as a kind of intellectual machismo. The style grew on me a bit while I was reading it, but on the whole it's not a friendly book.
Still � wax on, wax off. I read it. There's probably a better book to learn analysis from, but then again, maybe the difficulty is helpful. I don't know. In any case, if you can get through this one, you'll probably learn something.
Principles of Mathematical Analysis is the Old Master. It is completely uncompromising � no diagrams, the proofs are often opaque, the definitions unmotivated � and the book carries more than a whiff of that sadistic strain in math education that sees formal rigor and a lack of justification as a kind of intellectual machismo. The style grew on me a bit while I was reading it, but on the whole it's not a friendly book.
Still � wax on, wax off. I read it. There's probably a better book to learn analysis from, but then again, maybe the difficulty is helpful. I don't know. In any case, if you can get through this one, you'll probably learn something.
November 3, 2021
There are many things to like and dislike about this infamous textbook on mathematical analysis. Something to like is the clear exposition near the beginning. Something to dislike is the speed of coverage: going from the integers to Hilbert space in a little over 330 pages. I agree with other reviewers who find the first and second thirds of the book to be stronger than the final third, the latter which if it were part of a lecture course would surely be given the disclaimer 'non-examinable'. The author's recommendation that the text may be used for first-year mathematics lecture courses is now, as it must have also been in the 1950s and 1960s, somewhat unrealistic.
August 8, 2022
I think the most important thing to say about this textbook, in its favor, is: If you know the content of this book then you know what people expect you to know in real analysis. In that sense this books a good measuring stick of what you are supposed to know.
But the writing and pedagogy are absolutely terrible. This is just a result of the book being old.
Today, I think Terrance Tao has written a book that both covers the same material and then some, AND his pedagogical ability is perhaps the best of all mathematics authors for advanced subjects. If are choosing a book to learn from (rather than, say, as a reference text) I would strongly recommend Tao's Analysis 1 and 2.
But the writing and pedagogy are absolutely terrible. This is just a result of the book being old.
Today, I think Terrance Tao has written a book that both covers the same material and then some, AND his pedagogical ability is perhaps the best of all mathematics authors for advanced subjects. If are choosing a book to learn from (rather than, say, as a reference text) I would strongly recommend Tao's Analysis 1 and 2.
August 7, 2016
What else would one pick if s/he wanted to understand analysis in a higher level? Anyone who wants to begin studying analysis he could use this book to establish the necessary fundamentals in his head. Then he can go for other books (as I will go) to check their problems and find out about the structure they have used to think about various topics of this field. :)
August 23, 2009
Baby Rudin is one of the clearest introductory level analysis textbooks out there. The price might not be ideal (over a hundred dollars for a book the size of a DVD), but the directness of the proofs is fantastic. There is no fluff in this book.
February 26, 2013
Excellent approach from metric spaces at the very beginning. A good book that worth trying. Stuck at differential forms
April 24, 2020
The proofs are rather terse in a manner more obnoxious than elegant
November 1, 2023
A bit dry.
March 3, 2023
everyone’s gotta read the bible at least once right
December 29, 2019
This is the definitive book for real analysis.
The proofs here are notoriously terse; the first time I started this book I remember getting stuck on Rudin's proof of the fact that a set of rational numbers has no greatest element. And that's only the beginning. The format of the book is simply "Definition, Theorem, Proof" over and over; there is little to no exposition except when absolutely necessary. Thankfully, in the modern era there are a multitude of resources available for someone going through this book. It also helps to have an instructor/mentor that can motivate the material.
The actual calculus only starts in Chapters 5 & 6 (dealing with differentiation and integration respectively). Before, one has to start with learning what the set of real numbers are and why they are useful (covered in chapter 1). The two key consequences of the construction of the real numbers are the Archimedean Property and the denseness of the rational numbers in the reals, both of which become very handy later on. In addition to the field properties and discussion of least upper bounds and greatest lower bounds (sup and inf respectively), chapter 1 sets up the groundwork of introducing the extended real number systems, the complex field, and Euclidean spaces. There is a reason why the subject is called real analysis and not rational analysis.
Chapter 2 is where the abstraction goes up a notch. After a short discussion on countability, Rudin dives into the properties of metric spaces and their associated objects (open & closed sets, etc.). Here, the abstraction of distance is important. By focusing on the overall idea of a metric (distance), the concepts in chapter 2 can be extended not just to the real numbers but also to other sets such as C(X) (the set of continuous, bounded functions on a set X) and to other metrics such as the discrete and Lp metrics. A particular highlight of chapter 2 is Rudin’s coverage of compactness. By proving various theorems associated with compact sets (for instance, that compact subsets of metric spaces are closed), Rudin is able to quickly prove the Heine-Borel theorem (closed and bounded sets in R are compact).
Chapter 3 is about sequences and series. Here, Rudin formulates the concept of convergence of a sequence and (in a metric space) connects the properties of a metric space to sequences. Rudin then discusses other aspects of sequences including subsequences and Cauchy-ness. The extensive discussion of sequences are important due to the fact that infinite series can be seen as the limit of the sequence of partial sums (and so the properties of sequences carry over). The rest of the chapter covers various series tests and properties of series. Of particular note are Rudin’s discussions of the Ratio test and multiplication of series (which plays a role in chapter 8 when he develops the exponential function). Notably, Rudin chooses to discuss limsup and liminf in the context of the sup and inf respectively of the set of subsequential limits. Although his definition of limsup is useful for his proof of the ratio test, I would recommend looking at other formulations of these concepts.
Chapter 4, which discusses limits and continuity, connects the material in chapter 2 and 3 together. Instead of the typical epsilon-delta definition which one is often forced to learn, Rudin is able to formulate continuity in terms of neighborhoods and sequences. As a result, limits are easier to visualize and understand. Also, the ideas of compactness become important when Rudin introduces the idea of uniform continuity and the idea of a function having a max and a min on a closed and bounded interval on R. This particular idea becomes important for differentiation and integration. Rudin’s discussion of connectedness allows him to quickly prove the intermediate value theorem. Rudin also discusses limits at infinity. This idea becomes important for the proof of L’Hopital’s rule and other theorems.
Chapter 5 is all about derivatives. The mean value theorem is of particular importance since it allows one to examine how the derivative at a point affects a function’s behavior on an interval (given that certain hypotheses are fulfilled). Extending the mean value theorem gives Taylor’s theorem which allows one to connect a function’s derivatives to the function itself. Rudin also proves L’Hopital’s rule. This becomes an important computational tool in the later chapters.
Chapter 6 focuses on the theory of integration. Contrary to what is often believed in a typical first-year calculus course, discontinuous functions are also integrable. Therefore, Rudin starts from scratch by examining upper and lower sums in order to define what the integral of a function over a closed & bounded interval actually is. Rudin uses the Riemann-Stieltjes integral in his proofs. Although the proofs have an added layer of complexity (for example, showing that a continuous function is integrable), the payoff comes in both the ability to connect integrals and series and to give a short proof of the change of variables theorem for integrals. The proofs of the Fundamental Theorems of Calculus are quickly dispensed along with the integration by parts theorem. The ideas of the FTC are subtly used in later chapters. Both chapters 5 & 6 cover differentiation and integration of vector-valued functions respectively. Chapter 6 ends with a short discussion on rectifiable curves. The ideas in this section come up in Rudin’s discussion of the trigonometric functions.
Chapter 7 deals with sequences and series of functions. One could simply just extend the ideas of chapter 3 to pointwise convergence of functions. However, Rudin shows in the beginning of this chapter that problems arise when limit interchanges are done carelessly. As such, the stronger notion of uniform convergence is needed. After discussing uniform convergence and proving some theorems involving the Cauchy criterion and the M-test for series of uniformly convergent functions, Rudin quickly covers uniform convergence and limits, integration, and differentiation. Rudin then uses the notion of uniform convergence to construct a continuous function that is nowhere differentiable. The rest of chapter 7 deals with addressing the question of whether every bounded sequence of functions contains a convergent subsequence of functions. This question is not a simple as it may seem, and the additional property of equicontinuity is developed (extending the idea of uniform continuity to all functions in family). This discussion leads to the Arzela-Ascoli theorem (theorem 7.25). The rest of the chapter focuses on establishing and extending the idea that a continuous function on a compact interval can be uniformly approximated by a sequence of polynomials. After discussing function algebras, Rudin ends the chapter with proofs of the Stone-Weierstrass theorem in the spaces of real and complex continuous functions. The crucial ideas of the Stone-Weierstrass theorem come up in chapter 8 when Rudin asserts that a trigonometric polynomial can approximate a continuous, periodic function.
Chapter 8 focuses on bundling together the concepts of chapters 1-7 to prove claims about common functions. After a short discussion on power series that establishes ideas such as Taylor series and interchange of summation, Rudin focuses on constructing the exp, log, and trig functions. The groundwork in the previous chapters is applied to rigorously prove previously known properties. Rudin includes a short summary of Fourier series and includes a short proof of the Fundamental Theorem of Algebra. The chapter ends with a discussion of the gamma and beta functions which feature heavily in probability and other areas.
The book is difficult to complete but also hugely rewarding. The material combined with the difficult exercises can make the book tough to follow. Yet, the book still remains a cornerstone of real analysis classes and an excellent starting point.
The proofs here are notoriously terse; the first time I started this book I remember getting stuck on Rudin's proof of the fact that a set of rational numbers has no greatest element. And that's only the beginning. The format of the book is simply "Definition, Theorem, Proof" over and over; there is little to no exposition except when absolutely necessary. Thankfully, in the modern era there are a multitude of resources available for someone going through this book. It also helps to have an instructor/mentor that can motivate the material.
The actual calculus only starts in Chapters 5 & 6 (dealing with differentiation and integration respectively). Before, one has to start with learning what the set of real numbers are and why they are useful (covered in chapter 1). The two key consequences of the construction of the real numbers are the Archimedean Property and the denseness of the rational numbers in the reals, both of which become very handy later on. In addition to the field properties and discussion of least upper bounds and greatest lower bounds (sup and inf respectively), chapter 1 sets up the groundwork of introducing the extended real number systems, the complex field, and Euclidean spaces. There is a reason why the subject is called real analysis and not rational analysis.
Chapter 2 is where the abstraction goes up a notch. After a short discussion on countability, Rudin dives into the properties of metric spaces and their associated objects (open & closed sets, etc.). Here, the abstraction of distance is important. By focusing on the overall idea of a metric (distance), the concepts in chapter 2 can be extended not just to the real numbers but also to other sets such as C(X) (the set of continuous, bounded functions on a set X) and to other metrics such as the discrete and Lp metrics. A particular highlight of chapter 2 is Rudin’s coverage of compactness. By proving various theorems associated with compact sets (for instance, that compact subsets of metric spaces are closed), Rudin is able to quickly prove the Heine-Borel theorem (closed and bounded sets in R are compact).
Chapter 3 is about sequences and series. Here, Rudin formulates the concept of convergence of a sequence and (in a metric space) connects the properties of a metric space to sequences. Rudin then discusses other aspects of sequences including subsequences and Cauchy-ness. The extensive discussion of sequences are important due to the fact that infinite series can be seen as the limit of the sequence of partial sums (and so the properties of sequences carry over). The rest of the chapter covers various series tests and properties of series. Of particular note are Rudin’s discussions of the Ratio test and multiplication of series (which plays a role in chapter 8 when he develops the exponential function). Notably, Rudin chooses to discuss limsup and liminf in the context of the sup and inf respectively of the set of subsequential limits. Although his definition of limsup is useful for his proof of the ratio test, I would recommend looking at other formulations of these concepts.
Chapter 4, which discusses limits and continuity, connects the material in chapter 2 and 3 together. Instead of the typical epsilon-delta definition which one is often forced to learn, Rudin is able to formulate continuity in terms of neighborhoods and sequences. As a result, limits are easier to visualize and understand. Also, the ideas of compactness become important when Rudin introduces the idea of uniform continuity and the idea of a function having a max and a min on a closed and bounded interval on R. This particular idea becomes important for differentiation and integration. Rudin’s discussion of connectedness allows him to quickly prove the intermediate value theorem. Rudin also discusses limits at infinity. This idea becomes important for the proof of L’Hopital’s rule and other theorems.
Chapter 5 is all about derivatives. The mean value theorem is of particular importance since it allows one to examine how the derivative at a point affects a function’s behavior on an interval (given that certain hypotheses are fulfilled). Extending the mean value theorem gives Taylor’s theorem which allows one to connect a function’s derivatives to the function itself. Rudin also proves L’Hopital’s rule. This becomes an important computational tool in the later chapters.
Chapter 6 focuses on the theory of integration. Contrary to what is often believed in a typical first-year calculus course, discontinuous functions are also integrable. Therefore, Rudin starts from scratch by examining upper and lower sums in order to define what the integral of a function over a closed & bounded interval actually is. Rudin uses the Riemann-Stieltjes integral in his proofs. Although the proofs have an added layer of complexity (for example, showing that a continuous function is integrable), the payoff comes in both the ability to connect integrals and series and to give a short proof of the change of variables theorem for integrals. The proofs of the Fundamental Theorems of Calculus are quickly dispensed along with the integration by parts theorem. The ideas of the FTC are subtly used in later chapters. Both chapters 5 & 6 cover differentiation and integration of vector-valued functions respectively. Chapter 6 ends with a short discussion on rectifiable curves. The ideas in this section come up in Rudin’s discussion of the trigonometric functions.
Chapter 7 deals with sequences and series of functions. One could simply just extend the ideas of chapter 3 to pointwise convergence of functions. However, Rudin shows in the beginning of this chapter that problems arise when limit interchanges are done carelessly. As such, the stronger notion of uniform convergence is needed. After discussing uniform convergence and proving some theorems involving the Cauchy criterion and the M-test for series of uniformly convergent functions, Rudin quickly covers uniform convergence and limits, integration, and differentiation. Rudin then uses the notion of uniform convergence to construct a continuous function that is nowhere differentiable. The rest of chapter 7 deals with addressing the question of whether every bounded sequence of functions contains a convergent subsequence of functions. This question is not a simple as it may seem, and the additional property of equicontinuity is developed (extending the idea of uniform continuity to all functions in family). This discussion leads to the Arzela-Ascoli theorem (theorem 7.25). The rest of the chapter focuses on establishing and extending the idea that a continuous function on a compact interval can be uniformly approximated by a sequence of polynomials. After discussing function algebras, Rudin ends the chapter with proofs of the Stone-Weierstrass theorem in the spaces of real and complex continuous functions. The crucial ideas of the Stone-Weierstrass theorem come up in chapter 8 when Rudin asserts that a trigonometric polynomial can approximate a continuous, periodic function.
Chapter 8 focuses on bundling together the concepts of chapters 1-7 to prove claims about common functions. After a short discussion on power series that establishes ideas such as Taylor series and interchange of summation, Rudin focuses on constructing the exp, log, and trig functions. The groundwork in the previous chapters is applied to rigorously prove previously known properties. Rudin includes a short summary of Fourier series and includes a short proof of the Fundamental Theorem of Algebra. The chapter ends with a discussion of the gamma and beta functions which feature heavily in probability and other areas.
The book is difficult to complete but also hugely rewarding. The material combined with the difficult exercises can make the book tough to follow. Yet, the book still remains a cornerstone of real analysis classes and an excellent starting point.
October 27, 2019
I can't give it more or less than 3 stars as I only made it about halfway through before switching to Tao's Analysis I and II. I suspect this is a book loved by people already well versed in analysis, and/or those wishing to flaunt mathematical e-penii. Given the modern alternatives, would not recommend.
June 5, 2020
The classic analysis textbook. Hate it or love it, it is undeniable that this book has played a profound role in the education of budding mathematicians for the past several generations. It covers the material fairly well, and has a wide selection of exercises, ranging from routine to incredibly difficult.
Critics argue that the exposition is dry, terse and often unmotivated. While I can sympathize with this to an extent, I think this concern is overblown. The terseness means that your brain has to work harder to keep up. This is a good thing, not a bad thing. Rudin wrote the book in the style of a mathematician. His writing is not patronizing or "dumbed down". The book challenges students. It's the sort of book that can really bring out one's true potential. The book remains clear, precise and largely focused throughout, despite the dryness.
My own criticism is that the sections on Fourier analysis, multivariate analysis, differential forms and integration on manifolds were incredibly unsatisfactory and superficial. There are, in fact, some errors as well. For instance, in the section on Fourier series, there is a point where he uses two mutually contradictory definitions for the Fourier coefficients. This is easy to miss unless you pay close attention. It's not a serious issue, but given the terseness of the work, errors can easily throw a reader off.
I'd still recommend this book, for the exercises if nothing else. For multivariate analysis, I would strongly recommend Munkres' Analysis on Manifolds. One really needs a whole book on multivariate analysis; it cannot be a chapter or two.
Critics argue that the exposition is dry, terse and often unmotivated. While I can sympathize with this to an extent, I think this concern is overblown. The terseness means that your brain has to work harder to keep up. This is a good thing, not a bad thing. Rudin wrote the book in the style of a mathematician. His writing is not patronizing or "dumbed down". The book challenges students. It's the sort of book that can really bring out one's true potential. The book remains clear, precise and largely focused throughout, despite the dryness.
My own criticism is that the sections on Fourier analysis, multivariate analysis, differential forms and integration on manifolds were incredibly unsatisfactory and superficial. There are, in fact, some errors as well. For instance, in the section on Fourier series, there is a point where he uses two mutually contradictory definitions for the Fourier coefficients. This is easy to miss unless you pay close attention. It's not a serious issue, but given the terseness of the work, errors can easily throw a reader off.
I'd still recommend this book, for the exercises if nothing else. For multivariate analysis, I would strongly recommend Munkres' Analysis on Manifolds. One really needs a whole book on multivariate analysis; it cannot be a chapter or two.
September 3, 2018
The copy of Principles of Mathematical Analysis by Walter Rudin that I own is interesting in one way; it states that it is the Indian Edition. Now I don’t know much about publishing, but the biggest issue for me was whether or not the book was in English since I don’t know any Indian languages. I mean, I suppose the paper making up the book is slightly thinner and perhaps it uses a different measure of size, but other than that it didn’t need to say that on the cover.
This mathematical book is much like any other mathematical textbook that I own and have read, it starts with the basics and builds upon those basics in a systematic manner. The book contains proofs of theorems and practice problems, making it a very good resource.
I have heard that this book is used as a textbook in classes, but I never had to take a class in Analysis. As I might have mentioned long ago, all of the books that I read are only for my own amusement. However, it would be neat if I also learned something along the way.
The book delivers in being amusing and informative. I suppose it might be less amusing if this was a book I was assigned, but that is beside the point. In being informative, the book contains eleven chapters and covers subjects from The Real and Complex Number Systems to Lebesgue Theory. Finally, the book has a bibliography, an index, and a list of the special symbols used in the book.
This mathematical book is much like any other mathematical textbook that I own and have read, it starts with the basics and builds upon those basics in a systematic manner. The book contains proofs of theorems and practice problems, making it a very good resource.
I have heard that this book is used as a textbook in classes, but I never had to take a class in Analysis. As I might have mentioned long ago, all of the books that I read are only for my own amusement. However, it would be neat if I also learned something along the way.
The book delivers in being amusing and informative. I suppose it might be less amusing if this was a book I was assigned, but that is beside the point. In being informative, the book contains eleven chapters and covers subjects from The Real and Complex Number Systems to Lebesgue Theory. Finally, the book has a bibliography, an index, and a list of the special symbols used in the book.
April 13, 2023
A useless book if you are a beginner who has never seen a proof, and even more useless to learn on your own. And yet, it’s praised as a masterpiece in the math library. Why? In my humble opinion, it’s due to
1) Raw inertia. Everyone praises it, therefore, if you don’t bend the knee to Rudin-sama’s terseness, you’re an outlier. The same way that few think about criticizing a holly cow like Citizen’s Kane.
2) Sheer elitism. That’s it. Math communities tend to be elitist gatekeepers. As a result, they believe that presenting the material in the most opaque way is the best way to “learn� math.
Speaking now in math jargon, Baby Rudin is isomorphic to a dictionary. A collection of indexed Defiitions, Theorems and Proofs with an occasional remark inserted in-between, and presenting arguments that may seem to be taken out of the blue unless you are already versed in the subject. As such, this thing is a competent reference, but, in the same wat no one would try to learn a foreign language grabbing a dictionary and going word by word, I wouldn’t recommend to anyone trying to “learn� from this irredeemable piece. The only redeemable quality are its exercises, highly instructive and iluminating, but they are useless if you’re left behind from a subpar explanation (or lack of). As an alternative, go for Pugh’s Real Mathematical Analysis, Tao’s both volumes or Carothers. Stay away from this crap.
1) Raw inertia. Everyone praises it, therefore, if you don’t bend the knee to Rudin-sama’s terseness, you’re an outlier. The same way that few think about criticizing a holly cow like Citizen’s Kane.
2) Sheer elitism. That’s it. Math communities tend to be elitist gatekeepers. As a result, they believe that presenting the material in the most opaque way is the best way to “learn� math.
Speaking now in math jargon, Baby Rudin is isomorphic to a dictionary. A collection of indexed Defiitions, Theorems and Proofs with an occasional remark inserted in-between, and presenting arguments that may seem to be taken out of the blue unless you are already versed in the subject. As such, this thing is a competent reference, but, in the same wat no one would try to learn a foreign language grabbing a dictionary and going word by word, I wouldn’t recommend to anyone trying to “learn� from this irredeemable piece. The only redeemable quality are its exercises, highly instructive and iluminating, but they are useless if you’re left behind from a subpar explanation (or lack of). As an alternative, go for Pugh’s Real Mathematical Analysis, Tao’s both volumes or Carothers. Stay away from this crap.
January 24, 2020
Any undergraduate student will love to hate this book. Baby Rudin is a frustrating book due to its brevity. The topics covered however will not disappoint. It is only after you've finished the book you realize the depth of topics covered. Prerequisites include a basic understanding of set theory and operations, and a high school level calculus proficiency.
This text is a classic for anyone trying to learn real analysis. May not be suitable for self-learning if you're on a schedule.
This book will take you through
number systems and their construction, basic topological concepts of finiteness, compactness, closure, openness, finiteness and infiniteness of sets, sequences, series, continuity, differentiability, Riemann and Stieltjes integrals, and finally, Lebesgue theory.
This text is a classic for anyone trying to learn real analysis. May not be suitable for self-learning if you're on a schedule.
This book will take you through
number systems and their construction, basic topological concepts of finiteness, compactness, closure, openness, finiteness and infiniteness of sets, sequences, series, continuity, differentiability, Riemann and Stieltjes integrals, and finally, Lebesgue theory.
May 11, 2015
Rudin's Principles of Mathematical Analysis provides a pretty decent grounding in Real Analysis. Anyone who does anything with calculus should probably read it.
That said, it isn't a perfect primer. The proofs can be difficult to follow, and the language is very high-level. Some chapters suffer from a lack of examples or explanation. To get the most out of this book, it really has to be a classroom companion; you're not going to get too much out of just reading it in your spare time.
June 5, 2021
Classic presentation of real analysis for undergrad Math and graduate quantitative social science.
Very good for the basic topology, metric spaces concepts, calculus. I think this serves as a de facto standard for this at least in North America.
Measure spaces and Lebesgue integration are better discussed in Kolmogorov and Fomin's Intro to real analysis, which also covers more advanced linear algebra and functional analysis concepts.
Very good for the basic topology, metric spaces concepts, calculus. I think this serves as a de facto standard for this at least in North America.
Measure spaces and Lebesgue integration are better discussed in Kolmogorov and Fomin's Intro to real analysis, which also covers more advanced linear algebra and functional analysis concepts.
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