Reviews (73)

An excellent textbook
I think mathematics is a part of our culture.That's why, as a non-math major, I wandered into a very serious analysis class for mathematics majors.That might have been a disaster for me.Luckily, we used this book as a text, and it saved me.I read the whole book and diligently did all the exercises (of course, back then, it was the first edition, with only 227 pages and 140 exercises; it's somewhat more now).And that is my recommendation today.Read the book carefully and do as many exercises as you can.It certainly isn't easy.But it isn't, um, countably hard either.

The material in the book is self-contained.I guess that in theory, it could be mastered by any bright 14-year old who had learned some high school algebra and geometry.But I would surely recommend having much more mathematical sophistication than that as a prerequisite!

If you haven't learned the language of mathematics before, you'll enjoy the use of terms such as "countable," "real," "rational cuts," "measure," "ring," and "complete." By the end of the book, when the author claims that a proof (involving Cauchy sequences no less) is complete, you'll barely be able to suppress a desire to ask "Does every Cauchy sequence in the proof converge?"

In the first edition of this book, Rudin did mess up a little in his section on "the integral as a limit of sums." His theorem as stated was false.We cruelly dubbed it "Rudin's Last Theorem."Worse, he had used it "to prove some elementary properties of the Stieltjes integral."But that was all straightened out by the second edition.

I especially like the first couple of chapters.They give most readers the confidence to continue.And the final chapter, on Lebesgue integration, is very well written.One note of warning, though.Rudin begins this chapter by saying, "Some of the easier propositions are stated without proof.However the reader who has become familiar with the techniques used in the preceding chapters will certainly find no difficulty in supplying the missing steps."That is an exaggeration.It takes work.After all, this is, um, real mathematics you'll be doing!

I'm thankful that I was assigned this as my textbook.

A masterpiece
I absolutely agree with Professor Jorgensen.

I loved it when I was a student of physics, and I still love it because I tend to consider it as my personal standard in Classical Mathematical Analysis (and not only): sort of a "pacemaker" which sets the qualitative level to go back to just every time one is a little confused about what to do and where to go ;)

Reviews (17)

Standard Analysis Text; Necessary, but not Sufficient!
Very useful textbook, easy to follow.
A little over-simplified though.
Doesnot leave a lot of gaps, but then that doesnot leave a lot of room for imagination either.
At times it gets toooo wordy, if you know what I mean. Less ideas and more talk.
A stark contrast to Rudin, but can be very useful if both are used together. However, if you need to choose between those two, go with Rudin.

this book is just plain good.
I began as a graduate student in applied maths less than a year ago; all of the students that I spoke with prior to that said that real analysis with rudin's book was their worse & hardest class..
So when I walked into MTH 5111 Real Variables I thought oh *&^% what am I in for?? but then I picked up the Royden book and I understood the way he was presenting the materail.. the book is very stright to the point + leaves channelgning problems to the HW sets but the autor clearly outlines. I have learned more from this book and course than any other...

Reviews (16)

A start in math.
I am a fan of Rudin's books. This one "Real and Complex Analysis" has served as a standard textbook in the first graduate course in analysis at lots of universities in the US, and around the world.

The book is divided in the two main parts, real and complex analysis. But in addition, it contains a good amount of functional and harmonic analysis; and a little operator theory.

I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.

What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.

Some of the exercises are notoriously hard, but I think that is good: It simply means that they serve as work-projects when the students use the book. And this approach probably is more pedagogical as well.

After surviving some of the hard exercises in Rudin's Real and Complex, I think we learn things that stay with us for life; you will be "marked for life!"

Review by Palle Jorgensen, September 2004.

Welcome to the self-service analysis center!
This year we have been using Walter Rudin's treatise as the main text for a standard first-year graduate sequence on real analysis, backed up by Wheeden/Zygmund's book on Measure and Integral, and the two seem to complement each other quite nicely. Rudin writes in a very user-friendly yet concise manner, and at the same time he masterfully manages to maintain the high level of formality required from a graduate mathematics text. To be totally honest, a few years ago my very first attempt at learning graduate-level real analysis in a classroom setting (via Folland's book) was not successful, as I found the exposition in Folland very dense and rigid, and the homework problems too difficult to do. Rudin's book however is a lot more accessible for the beginning graduate students who may not have had any more than some basic exposure to measure theory in their upper division analysis classes. One point to keep in mind is that Rudin developes the measure in the more formal axiomatic way, instead of in the more concrete constructive approach. In the constructive approach, one first introduces the "subadditive" outer measure as a set function which is defined on the power set P(X) of a nonempty set X. One then proceeds by showing that the restriction of the domain of the outer measure to a smaller class of subsets of X (a sigma algebra M), obtained via applying the Caratheodory's criterion, results in a "countably additive" set function which is called a measure on (X,M). (The latter is the approach taken in both H.L. Royden and Wheden/Zygmund). The formal approach is not very intuitive and is less natural for a beginning graduate student who might not have developed a certain level of mathematical maturity yet.

Also, Rudin does not discuss some of the more advanced or interdisciplinary topics such as distribution theory (Sobolev spaces, weak derivatives, etc.) or applications of measure theory to the probability theory, both explored in the book by Folland. Last but not least, it's worth noting that contrary to the common practice, Folland includes many end-of-chapter notes where he outlines some important historical aspects of the development of the topics, and also gives a few references for further study. For example, in the notes section at the end of the chapter on Lebesgue integration, he mentions --and briefly outlines-- the basics of the theory of "gauge integration" (also called Henstock-Kurzweil theory) which serves to construct a more powerful integral than that of the Lebesgue's. As another instance, having already defined and used "nets" within the chapter on topology, in the end-notes Folland also introduces "filters" and "ultrafilters". These are all machineries which have been developed to play the role of the metric space sequences in general (locally Hausdorff) topological spaces, but for some historical reasons, ultrafilters have nowadays taken a backseat to the nets (the older general topology books usually prove the Tychonoff theorem using ultrafilters). All said, I can recommend taking up Royden as your very first approach to measure theory, then based on how well you think you have learned the first course, move on to either Rudin or Folland for a more advanced treatment. Please note that the other books I have mentioned above do not discuss complex analysis, a subject which is also masterfully presented in Rudin. There are however a few other equally well-written complex analysis books to pick from, for example John B. Conway's classic from the Springer-Verlag graduate series, or L.V. Ahlfors' masterpiece, to name just a couple.

A Comprehensive Guide to Analysis
Rudin's Real and Complex Analysis is an excellent book for several reasons.Most importantly, it manages to encompass a whole range of mathematics in one reasonably-sized volume.Furthermore, its problems are not mere extensions of the proofs given in the text or trivial applications of the results- many of the results are alternate proofs to major theorems or different theorems not proved.With that in mind, this book is not appropriate for a course where the instructor wants students to merely understand the theorems well enough to develop applications- the structure of the book is far better suited for a more theoretical course.

For example, the construction of Lebesgue measure is considered one of the most important topics in graduate analysis courses.After this construction, more abstract measures are developed, and then one proves the Riesz Representation Theorem for positive functionals later.

Conversely, Rudin develops a few basic topological tools, such as Urysohn's Theorem and a finite partition of unity, to construct the Radon measure needed in a sweeping proof of Riesz's Theorem.From this, results about regularity follow clearly, and the construction of Lebesgue measure involves little more than a routine check of its invariance properties.

Another example of where Rudin takes a more theoretical approach to provide a more elegant, yet less intuitive proof, is the Lebesgue-Radon-Nikodym theorem.Other books generally introduce signed measures with several examples, and use this result, along with properties of measures to derive the proof.On the other hand, since the first half of the book contains an intermission on Hilbert Space, Rudin uses the completeless of L^2 and the Riesz Representation Theorem for a more sweeping proof.

In the real analysis section, Rudin covers advanced topics generally not covered in a first course on measure theory.The chapters on differentiation and Fourier analysis are key examples of this.Rudin uses maximal functions to develop the Lebesgue Point theorem and results from complex analysis, and provides an incredibly thorough proof of the change-of-variables theorem.The ninth chapter, on Fourier transforms, relies heavily on convolutions, which are developed as a product of Fubini's theorem.This, in turn, is used to prove Plancherel's theorem and the uniqueness of Fourier transforms as a character homomorphism.

The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane.Once a solid foundation on the topic is laid, Rudin can develop more advanced topics from Harmonic analysis using general results from real analysis like the Hahn-Banach theorem and the Lebesgue Point theorem (for Poisson integrals).

Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition.Nonetheless, all the basic results are covered, and Rudin uses these to develop the main theorems, such as the Mittag-Leffler and Weierstrass theorems on meromorphic functions, and the Monodromy Theorem and a modular function used to prove Picard's Little Theorem.

Reviews (4)

Decent book, if you can get it cheap
I strongly urge any serious math student to own a copy of both Rudin's Principles ("Baby Rudin") and his Real and Complex Analysis ("Adult Rudin").The former is absolutely essential- without completely mastering continuity and convergence on the basic metric space topology on R^n, higher math is going to be quite a pain.The second is good because it puts the major ideas of basic analysis- Radon measures, L^p spaces, rudiments of Hilbert and Banach Spaces, differentiation and integration, Fourier and Harmonic Analysis, Holomorphic and meromorphic functions, etc. all in one nice volume, although the problems may be too challenging or tangential to master the material by doing them.

With that said, I don't like this book as much.Perhaps because the problems don't provide great movitation for the theorems- in any event, I would recommend using at least two books to understand functional analysis.One that emphasizes a rigorous approach to the theory involved, and another more applied book that allows you to play with the new tools to solve the problems functional analysis was invented to solve; quantum mechanics, for example.

Reed and Simon is a good book, although I'm sure physicists or physics students would probably complain about it for the same reason I like it- its very mathematically rigorous and has a ton of problems- 30 to 60 on average at the end of each chapter, with only a few digressions into applications into quantum physics or elementary QFT.Get this with some Springer text, like Elements of Functional Analysis.

One more note- Rudin's book is broken up into three parts- one on TVS (Topological vector spaces) that combines topological properties of a space (for example, local convexity or local compactness) with the usual vector-space operations to set the spaces where operators act.

The second section deals with distributions- I regret that one failure of "Adult Rudin" was to emphasize the abstract integral as a linear functional, because this would have helped to make the concept of a distribution more clear.

While the introduction to distributions and their connections to Fourier analysis and differential equations is nice, the text gets bogged down with proofs about convolutions that are highly technical (and make either good practice or a good time for Rudin to actually use, for once, "The details are left to the reader...").

Finally, Rudin introduces operator theory, although it could go much more smoothly- the proofs come off as way too technical, a far cry from the "slickness" his proofs are often accused of being in the graduate analysis text.

All in all, there's some interesting problems to do, but you're not going to understand the applications of Functional Analysis to quantum mechanics or PDE (other than distributions a little), where other, more applied (read: easier) books may give nice problems about applications of Hilbert space methods, such as variational techniques or Fredholm theory.

Modern topics in math.
"Modern analysis" used to be a popular name for the subject of this lovely book. It is as important as ever, but perhaps less "modern". The subject of functional analysis, while fundamental and central in the landscape of mathematics, really started with seminal theorems due to Banach, Hilbert, von Neumann, Herglotz, Hausdorff, Friedrichs, Steinhouse,...and many other of, the perhaps less well known, founding fathers, in Central Europe (at the time), in the period between the two World Wars. In the beginning it generated awe in its ability
to provide elegant proofs of classical theorems that otherwise were thought to be both technical and difficult. The beautiful idea that makes it all clear as daylight: Wiener's theorem on absolutely convergent(AC) Fourier series of 1/f if you can divide, and if f has the AC Fourier series, is a case in point. The new subject gained from there because of its many sucess stories,- in proving new theorems, in unifying old ones, in offering a framework for quantum theory, for dynamical systems, and for partial differential equations. And offering a language that facilitated interdisiplinary work in science! The Journal of Functional Analysis, starting in the 1960ties, broadened the subject, reaching almost all branches of science, and finding functional analytic flavor in theories surprisingly far from the original roots of the subject. The topics in Rudin's book are inspired by harmonic analysis. The later part offers one of the most elegant compact treatment of the theory of operators in Hilbert space, I can think of. Its approach to unbounded operators is lovely.

Reviews (22)

Excellent intro. to complex analysis!
This course was my first exposure to the mathematical field of analysis at the undergraduate level, and our school ditched Gamelin's book used two years ago in favor of this book.Just to give you an idea of the difference a book makes (it was the same teacher for both courses, mind you):when Gamelin was used, EVERYONE dropped out of the course; when Brown/Churchill was used, only one person dropped the course and half the class received A's!

Truly, this is a remarkable shift, and this book had a lot to do with it.I thought the organization was flawless (note:you will have to go through the book in order, as many examples depend on previous material), and starting from the beginning with the definition of a complex number was definitely the way to go, as about 1/3 of my class had never seen a complex number before.I loved the fact that there were many examples worked out (never explicitly showing people how to do the end-of-section exercises, but showing them the methods for where to go) and the major theorems were alloted many pages for clear proofs with diagrams and detailed explanations (an entire section was devoted to a proof of the Cauchy-Goursat theorem!).Also, the choices of problems were superb, with some routine exercises meant to get you thinking along the right tracks followed by some very difficult ones.Basically, enough to challenge even the ablest math student, but enough for the average one to get a grasp on the concepts as well.

The book also provides an advantage for the instructor as to what applications to teach.Granted, chapters 1-6 cover almost all the theory, but 7-12 are all applications (7 is "usually" considered theoretical as well, but it is called "applications of residues!") in physics, advanced calculus and geometry, and engineering.So, a professor could choose to emphasize only the theoretical parts and save the apps. for independent study (which my prof. did) or could teach the relevant theories coupled with some of the applications (conformal mapping with fluid flow and heat flow, for example).It truly is a versatile book.

I noticed a complaint on here about not having enough examples or worked-out proofs.Well, to that individual (and any others who might be having the same problem), this book is meant for an upper-level undergraduate course, which means that there are going to be less examples worked out in great detail, the proofs may just be thumbnail sketches, and the problems will not have a quick reference page in the chapter for a formula or method like in calculus, for example; even though the book is versatile, a lot of the learning still falls on the student's shoulders.

My one and only gripe is that the book didn't take a lot of time to spell out how to perform a delta-epsilon proof for limits, which is one of the basic proofs in analysis.But, luckily, I had a very patient instructor who was willing to walk it through with me (most of the rest of the class had already had real analysis, so they didn't need to go over it).But, still, it's not enough to take it down a star, in my opinion.

They say this book is among the canon of undergraduate mathematics, and I can certainly see why.What a great introduction to complex analysis!This book will definitely be accompanying me to grad school!

excellent! (as an intro)
This book would be really good for a physics or engineering student, or for a math student as an intro since it isn't as theoretical as other books. Everything is explained very clearly, with lots of examples but still with some hard/interesting problems also. The first 7 chapters cover all the theory, which is the usual stuff like complex numbers, differentiability & Cauchy-Riemann equations, integration & Cauchy integral forumla/theorem, series, Taylor series, Laurent series, residues, poles, solving real integrals with residues, etc etc. Chapters 8-12 cover applications which I admit I don't know a lot about, but if the 1st half of the book is any indication, they are good also. The text covers everything in a CONCRETE way, as opposed to an abstract way using concepts from topology like connectedness, compactness, open/closed sets, star-shaped sets, etc. For that stuff I used GJO Jameson's Complex Functions text & I also like the one by Ahlfors & the one by Jerrold Marsden. I think every scientist or engineer would find this Brown/Churchill book very helpful. :-)

Wonderful intro to complex analysis.
I used this text for an undergrad course, and I thoroughly enjoyed it.The book, as the title suggests, is ideal for physicists, engineers, and applied mathematicians, in that it hits the important and powerful features of complex analysis which are tremendously useful for applied work.What's more, it succeeds in doing so without sacrificing mathematical rigor, and without getting bogged down on stodgy formalism.

As for the scope of the book, I believe it can be fairly stated that just about everything in the book should be studied and mastered by readers doing applied work.From my own experience, everything covered in the book has turned out to be relevant at one point or another.It can be said without exaggeration that this book is a gold mine.