Reviews (16)

Excellent Moore Method Intro Text
I used this text as senior undergraduate in an introductory course to real variables. The course was structured as a sort of modified RL Moore method class: there were very few lectures, and we (the students) could only use theorems and propositions presented in the text if we had gone to the board and presented a valid proof for each. As such, most of the students learned the fundamentals very well. This in turn made my first graduate course in real variables much easier.

The biggest downside however is that most graduate students don't have the time needed to dedicate to the various problems in this text, which is why Royden is probably not the best choice for a first year graduate text. Instead I would recommend Bartle's Elements of Integration and Lebesgue Measure as a first year grad text on the subject. It was disappointing to use Bartle and discover that so many of the problems in Royden, which I had spent countless hours attempting to prove, had been completely worked out in Elements of Integration.

In short, Royden makes you work for many (most?) important results, and in the long run this makes for a much stronger understanding of the material- if you have the time to devote to it.

Not bad for self-study, excellent for reference
I used Royden (2nd edition) as a graduate student over 30 years ago, and have been away from real analysis pretty much ever since (not because of the book(!), but because of being in computers). I've taken a renewed interest in the subject (I'm a pretty random person) and have been surprised at how the material has come back to me, I think because of the readability of the text. It's true, Royden challenges the reader at every turn, but if one has acquired the level of mathematical maturity commensurate with strong interest in analysis, the challenges are appropriate, in my opinion

Reviews (5)

Frustrating
It seems that the higher up you go in the Mathematics curriculum, the poorer the books you meet. In my honest opinion, a book should help you learn and understand the material as quickly as possible. Otherwise, you might as well be given a list of definitions, stuck in a closed padded room and asked to come up with all the theorems by yourself. Unfortunately, there are too many graduate textbooks out there written by individuals who seem to have no desire to make the ideas they are trying to present as clear as possible. There's no educational philosophy. This book falls under that category. For example, this book is almost completely devoid of any examples. I don't know about you, but from example, is how I learn. I could go through this book much faster, if there were some decent examples. You can tell me a thousand times what a sigma algebra is, but if you don't give me some decent, worked-out examples which might tell me why tell me why I should learn it (other than because I'll fail the course), I'm going to forget the definition after 5 minutes. Secondly, it would help if there were more pictures. A picture is worth a thousand words. Third, some of the definitions are not worded as well as they should be. last night I spent ten minutes trying to figure out whether the definition for x-section Ex = {y in Y : (x,y) in E} meant that "for all x," or just "for some x?" It turned out it meant "for fixed x." But nowhere was that little tidbit of information written. Ten minutes may not sound like much, but if you have to read 10 pages before you get to pleasure of spending 10 hours with the homework problems, that translates into a lot of time you could spend doing other things if only this book were presented in a manner which would enable you to learn the material more efficiently. I give it two stars primarily because some of the homework problems aren't too bad. If you have a choice, have a look at Kolmogorov and Fomins book on Real analysis. It's not perfect, but the material in it is organized better. (It's not as DENSE) Plus it's a Dover book, and therefore much cheaper.

Could have been great
I speak as a graduate student in applied math. I really like this book but was bothered by its flaws. Nevertheless, with a good instructor, this text can make for a good learning experience.

Positives: The book is well organized. It builds in a reasonable way so that I could focus on the material in the book and develop my understanding as I went. The book is reasonably well contained. Outside of a reasonable level of basics (a BA or BS in math) the proofs and most of the problems use material developed earlier in the text. I found the book very interesting -- I especially liked the topics presented in the last few chapters.

Negatives: Lots of typos - the author's errata sheet is woefully incomplete. Too few expamples. Too condensed - sometimes to the point of incomprehensibility or even error. The contents of a whole course may be condensed in to a single chapter or even a single section.

Things to be aware of: You should be comfortable with advanced calculus, topology, set theory, and algebra (linear and modern). It also helps to have had some basic real analysis. I highly recommend that you've seen Fourier transforms, Dirac deltas (distributions), and continuous probability. You aren't going to learn these here - you're going to see how measure theory is applied to them.

Reviews (15)

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.

As an introductory text, even for advanced students, Rudin should probably be accompanied by more descriptive texts to develop better intuition. In fact, I would recommend Folland's Real Analysis and Ahlfors' Complex Analysis for self-study, because the problems are easier and one can learn better through those. With a good instructor, though, Rudin's text is concise and elegant enough to be both useful and enjoyable. It is also a good test to see how well one REALLY knows the subject.

Excellent, often intriguing treatment of the subject
The first part of this book is a very solid treatment of introductory graduate-level real analysis, covering measure theory, Banach and Hilbert spaces, and Fourier transforms. The second half, equally strong but often more innovative, is a detailed study of single-variable complex analysis, starting with the most basic properties of analytic functions and culminating with chapters on Hp spaces and holomorphic Fourier transforms. What makes this book unique is Rudin's use of 20th-century real analysis in his exposition of "classical" complex analysis; for example, he uses the Hahn-Banach and Riesz Representation theorems in his proof of Runge's theorem on approximation by rational functions. At times, the relationship circles back; for example, he combines work on zeroes of holomorphic functions with measure theory to prove a generalization of the Weierstrass approximation theorem which gives a simple necessary and sufficient condition for a subset S of the natural numbers to have the property that the span of {t^n:n in S} is dense in the space of continuous functions on the interval. All in all, in addition to being a very good standard textbook, Real and Complex Analysis is at times a fascinating journey through the relationships between the branches of analysis.

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.

Reviews (1)

Isbn: 0387944605
Sales Rank: 383,065
Subjects:  1. Functions Of Complex Variables    2. Mathematical Analysis    3. Mathematics    4. Science/Mathematics   

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 (47)

A great book
I'm not really sure why people dislike this book. In general, Strang writes some of the best mathematics texts out there, and this book is no exception. In fact, my only objection to this book is that it is too introductory, as is _Intro to Applied Mathematics_, but they aren't after all intended for graduate courses.

The one thing I really enjoy about his texts is that his writing, even on basic material, sometimes contains very deep insights into the underlying structure of the mathematics. Understanding a Strang textbook is the difference between actually _understanding_ mathematics and just being able to do it.

So I heartily endorse this book. If you want a higher-level text, I'd suggest Demmel's Applied Numerical Linear Algebra. But reading this book still wouldn't hurt.

Excellent development of the subject
It was some years ago when I first took a Linear Algebra course and fortunately the instructor choose this as the text book. I thought that Strang's explanations were superior to the lectures given in class and I could have easily gotten through the course by reading this book and taking the tests ! Just recently I have had to go back and relearn this subject and this book was almost joy to read (well O.K. it is a math book). Mr Strang has the rare ability of making this subject intuitively obvious and convincing. He starts from very a modest beginning of solving systems of linear equations and then makes the almost seamless transition to matrices and vectors by a change of perspective. As the subject is developed, Mr Strang makes every effort he can to clearly present the material as does not defer to the usual "left as a an excercise for the reader" device to evade explaining the subject. The narrative is lively and enthusiatic and sometimes even humorous and has lots of reminders of previously mentioned ideas to keep the text flowing. If more math books were written like this, then fewer people would complain about obscurity of this subject.

Insight not rigor
This book seems to have evoked a wide range of emotional responses - unusual for a technical book. I am an analyst at a major aerospace company and have found it to provide tremendous insight into the real world problems while providing the mathematical tools for a broad range of problems. If you are a mathematician looking for rigor, then you need to go elsewhere as this book provides a conversational approach versus a theorem-proof approach. If doing it for self-study, make sure and work the problems as it introduces you to new areas for further study.

Reviews (3)

Great book on optimal controls
I had the pleasure of having taken my control courses at Georgia Tech while the author, Frank Lewis was a professor there. He was probably one of the best professors in this area amongst a stellar group of control professors. He explanations and presentation of the topics were concise, clear, and extremely applicable. These same traits are present in this book. One of the main knocks against control theorists is that they tend to refrain from using real world examples to illustrate their point. Dr. Lewis is quite the opposite, he believes that the real world problem is key to understanding the theory and he is never shy in using those examples. The number of examples may be a problem here, afterall, students live and die by the worked problem to get to understanding.

This book is an excellent introductory guide to the fascinating world of optimal controls.

a good book
This book is used as the textbook for my optimal control course.i think that the authors already done a good job in presenting the ideas.actually i understand the ideas pretty well through the help of this book.but again there is a lack of examples.sometimes i find difficulties in solving the problems as the illustrated examples are not so structured.nevertheless,it's a book for graduate students,so i think that it's ok.also it can serve as a good reference book

Reviews (1)

Reviews (4)

Very Good Book to Study Adaptive Control
I found this book explains the field of adaptive control very well. It covers its basic theory, its proof, and also its application. It is very good book for anybody who has interest in adaptive contro, and for anybody who will just begin to study about adaptive control.

Best First Book on Subject
One of the best introductory books on the subjects. I've been able to implement real adaptive controllers using this book. Particularly good for sliding-mode controllers, variable-structure controllers and adaptive PID controllers.