Reviews (4)

This is a book for REAL mathematicians
This book is an wonderful introduction to Differential Geometry for the serious student of mathematics. However, it is not aimed at engineers, physicists or even applied mathaticians.
The author assumes the reader has an extensive knowledge of abstract algebra and at least one course in analysis. Likewise, he has chosen to emphasis applications of the subject to Lie Groups, homotopy theory, and group actions, rather than the physical applications that applied mathematicians are looking for. But, for the student of pure mathematics, this text is a great starting point into the rich world of differential geometry.
Also, while this book is an introduction and requires no previous knowledge of the subject, it covers enough ground to be followed up by such topics as the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, or Morse Theory.

When accountants and soldiers take interest in geometry.....
One day, accountants and soldiers may take an interest in differential geometry. If and when such a day comes to pass, this book will have a role to play. Until then, engineers, physicists and mathematicians alike have better alternatives, such as the inspiring texts, with complementary qualities, by Burke, "Applied Differential Geometry"; by do Carmo, "Riemannian Geometry", or by Spivak, "A Comprehensive Introduction to Differential Geometry".

Even more advanced books such as Lang's or Petersen's are more readable: in them the extra formalism brings the reward of more powerful results. Here the retentive attention to the trees at the expense of the forest is merely a barrier to entry for the uninitiated. This text's popularity in some areas of engineering must have played a role in the slow acceptance of Riemannian geometric methods.

Manuel Tenide

Reviews (1)

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 (16)

why 4 stars and not five?
I was a (French)graduate student in France some 25 years ago and I would have been delighted to use this book if translated in French; I had to rely on Cartan's book which is a very good book too but which takes for granted that one already knows quite a lot on complex numbers, series, convergence and topology...As a substitute to Cartan, there was a translation of Rudin's real and complex analysis which begins with measure theory...Anyway, it is very difficult to learn this subject in any book without advice from instructors and attending lectures.
There could be more worked examples in this book but it is nota self teaching book (neither is Cartan's...which is very similar in essence to Ahlforsbut more narrow minded). For a more "basic" book in the subject, see Marsden's Basic complex analysis but proofs are often mixed up with exercises...which does not suit everybody. My final point is the following: this book contains much more stuff to work at or to think about than its French counterpart; moreover,in this book, efforts are made to avoid formalism (Bourbaki?). US maths students are very lucky indeed. But the book is certainly too expensive.

A Classic
Another classic text from graduate school (text for class taught by P.L. Duren) providing a background in introductory complex analysis.This book is nicely written with some elegant exploration of the motivations and backgound for a number of the central concepts.This may be surprising given the physical slimness of the text (I noticed elegance of the exposition and attention to motivation on a recent reread of some of the book after nearly twenty years -- I had not remembered this exposition, perhaps because the reading in graduate school was not quite as "liesurely" (unless "fear driven" and "pressured" are synonyms for "liesurely").The theory topics are nicely covered -- if, however, you are an engineer looking for methods of calculating complex intgral there are other texts.

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 (1)