Reviews (20)

DO NOT USE for a first course in abstract algebra
The major problem I have with Artin is his stubborn reliance in teaching group theory via linear algebra. In my experience, undergraduate students encountering group theory for the first time have only a limited background in linear algebra, so using this text book forces students to cover two (albeit related) subjects at once. His discussion of rings and fields, while concise at times, is fine, but I find his coverage of group theory lacking. No discussion is give to semi-direct products, nilpotent groups, or solvable groups and related topics such as commutator subgroups.

The worst math book in my entire undergraduate career,
This is quite possibly the worst math book that I have ever had to read so far. Artin does a terrible job of explaining anything, sometimes he gets hung up on details, sometimes he skips over a lot of important aspects. The explanations, the illustration of the main ideas are meager or non-existent. To me, this book seems like a collection of theorems with very, very little explanation in between. Many proofs are only sketched, you really need to grind your way through each section. Be prepared to get stuck a lot if this is the textbook for your first algebra course, not because you're stupid, just because Artin can't explain the subject. Just half the time, you don't know what he's talking about because you're LEARNING algebra and not using the book as a reference of some sort, which to me it seems, Artin is assuming. I read the other reviews about how great this book is. Don't believe a word of it. Some people apparently enjoy struggling with other people's incompetence to explain things clearly. Obviously, if you master this book, you've probably learned a lot but that is only because there is a lot material, not because it was such a good book. And you ended up proving everything yourself, not because the book helps you with that, just because it does such a bad job of proving anything. I have spent so much time on my algebra coursework with so little return because this book was very lousy. By the way, if everyone else in your course seems to be getting it, don't worry, they're not, they just think they know it because they didn't read any of the proofs or don't know the big picture either.

Reviews (27)

The book after Herstein
I think I would only recommend this book to someone who has already had some exposure to algebra (or one especially gifted in mathematics). The beginning of the book is not too bad, but towards the end of Part I the pace quickens quite a bit. If you are willing to read over the text many times, and do all of the non-trivial exercises (there is an impressive olla podrida of algebra in them, most of which are the beginnings of some very deep ideas), then it should be a very rewarding experience. Namely because this is one of the most readable textbooks which covers everything from groups, rings, and fields to homological algebra and algebraic geometry. It is very rare to see this much material covered in one book, and for it to remain so structured (Rotman is an example of a book that covers a lot of material, but loses its structure somewhere).

Excellent book
I am surprised that this book has not got the 5 stars. It is very suitable for advanced undergraduates/first-year graduates. The book is full of examples; and the proofs are amazingly clear and succinct. The book introduces new concepts in the excercises long before the student encounters them in the sections.

This is a beautiful way to teach mathemtatics,--and indeed to learn it. The book is replete with examples that connect concepts from toplogy and real analysis with Algebra.

This book definitely deserves the 5 STARS.

Comprehensive book
I'm a graduate student in math. We used this book for the basic year-long abstract algebra sequence: group theory, chapters 1-4 and some of chapter 5; ring/field/galois theory chapters 7-9, 13-14. Some of my fellow students took a module theory course which was at least partially based off chapters 10 and (I think) 11. I'm sure more advanced courses could easily be based off chapters 15-end. Considering the cost of university books, I consider it very nice to buy one book for essentially 3+ courses.

The exercises in some sections are very diverse. My group theory professor made us do a huge number of them, and now I am amazed at how often I see questions similar to those from Dummit-Foote show up on past qualifier exams from many different universities. Regarding lack of answers in the back...well, you shouldn't need too many, and if you get really stuck, that's what the professor is for. And if you're learning it on your own then I'm thinking you should be brainy enough not to need answers!

The text itself is very readable and complete.

I don't think I'd recommend this as an undergrad textbook, although I've no doubt that there are some clever undergrads who could learn from it. I used Herstein's "Topics in Algebra" for my intro-to-abstract course as an undergrad. Herstein is designed to be introductory in nature, though still a wonderful book, while DF is more encyclopedic.

Reviews (19)

Not really helpful
I tried to use this book as a supplement text for my multivariable calculus class. I found it to be very useless even as a supplement, let alone a main source to rely on. The main problem with this book is the fact that it is very short (Can I say way overpriced? Good thing I got it for free) and it doesn't explain the concepts properly. It is theoretical, but also in my opinion too far out there. I have nothing against theory, in fact I think its great to have theory in a math book. But in this case the material presented very tersely and unclearly. In my opinion books should explain concepts nicely and clearly with a proper use of examples. I do not wish to spend hours trying to understand what an author tried to say, especially when a concept is a really easy one. Another annoying thing about this book is the notation. Author uses "modern" notation for partial derivatives, but for some reason not many other people use it. It is found mostly in the 1950s era math books. This archaic approach to math is devastating to a student. Avoid at all costs.

Must be written by Spivak's evil twin
Spivak's other books are quite good, but don't let that fool you into getting this one. This is a horribly dry and terse text of the type which is convenient for authors and lecturers but hopeless to learn from. The object of Bourbakian worship is of course "the modern Stokes' Theorem", but, Spivak says in his preface, "Yet the proof of this theorem is, in the mathematician's sense, an utter triviality - a straight-forward computation. On the other hand, even the statement of this triviality cannot be understood without a horde of difficult definitions from Chapter 4. There are good reasons why the theorems should all be easy and the definitions hard." Perhaps these "good reasons" are that lazy authors can throw together unhelpful books where everything is "left to the reader".

Reviews (22)

Are all Silverman's "translations" like this one?
First, let us be precise in reviewing this book. It is NOT a book by Kolmogorov/Fomin, but rather an edited version by Silverman. So, if you read the first lines in the Editor's Preface, it states, "The present course is a freely revised and restyled version of ... the Russian original". Further down it continues, "...As in the other volumes of this series, I have not hesitated to make a number of pedagogical and mathematical improvements that occurred to me...". Read it as a big red warning flag. Alas, I would have to agree with the reader from Rio de Janeiro. I've been working through this book to rehash my knowledge of measure theory and Lebesgue integration as a prerequisite for stochastic calculus. And I've encountered many results of "mathematical improvements" that occurred to the esteemed "translator". Things are fine when topics/theorems are not too sophisticated (I guess not much room for "improvements"). Not so when you work through some more subtle proofs. Most mistakes I discovered are relatively easy to rectify (and I'm ignoring typos). But the latest is rather egregious. The proof of theorem 1 from ch. 9 (p.344-345) (about the Hahn decomposition induced on X by a signed measure F) contains such a blatant error, I am very hard pressed to believe it comes from the original. That book survived generations of math students at Moscow State, and believe me, they would go through each letter of the proofs. Astounded by such an obvious nonsense, I grabed the only other reference book on the subject I had at hand, "Measure Theory" by Halmos. The equivalent there is theorem A, sec. 29 (p.121 of Springer-Verlag edition), which has a correct proof.
For those interested in details, Silverman's proof states that positive integers are strictly ordered: k1Unfortunately, I don't have the Russian original. Instead, I'm trying to get the other, hopefully real translation, "Elements of the Theory of Functions and Functional Analysis". BTW, this is the actual title of the original, not "Introductory Real Analysis". Which apparently is causing significant confussion amoung past and present readers. To give you a background info, the Russian original is (or has been, at least) used as a textbook for a third-year subject for (hard-core) math students. Meaning, in the preceding two years they would complete a pre-requisite four-semester calculus course. For example, criteria of convergence of series and their properties is an assumed knowledge in presentation of Lebesgue integral. So, I think most of the critique from earlier reivews is a bit misdirected. The original book is a great starting book into functional analyis/Lebesgue integration and differentiation, but proofs require solid understanding of fundamentals of calculus.
The best part about Kolmogorov's text is the clarity of conceptual structure of the presented subject a reader would gain, if he/she puts some effort. You would gain a thorough understanding, not just a knowledge of the subject. There is quite a difference between the two, and not that many authors succeed in delivering that.
But to gain that from Kolmogorov, I would suggest the other, "unimproved" but real, translation.

Not so "introductory"
This textbook has several major virtues: it is dirt cheap, it is concise, and it touches on many advanced topics. Unfortunately, it has equally major flaws.

Many of the "proofs," especially in the first few chapters, are simply vague outlines of proofs. New notation is introduced without formal definition, terminology is used sloppily (sometimes even inaccurately), and explanations are invariably terse.

Before reading each chapter, I found it was necessary to first consult a more down-to-earth text. Sometimes I got the impression that the authors were more interested in showing off their brilliance than teaching me about analysis.

If you want to learn analysis, I would recommend first working through Rudin's Principles of Mathematical Analysis, then using this book as a source of challenging problems and interesting remarks.

Siverman spoiled Kolmogorv
This is one of many Dover's book where the editor Silverma tryed to ''improve'' the original version of a book.

Reviews (5)

perhaps the best introduction to complex analysis
This is the book that really made me understand basic complex analysis.It doesn't try to give the most sophisticated or slickest presentation for experts.Instead, it gives a beautiful, concrete, down to earth explanations.The best feature is the applications.D. J. Newman is one of the world's great problem solvers, and this book includes numerous examples of how to use complex analysis to solve problems in surprising ways.Even in the more standard applications, such as summing series, the book gives many unusual examples.It concludes with Newman's proof of the prime number theorem, which is substantially shorter and clearer than many other proofs.

Not enough for getting a complete perspective.
My comment refers to the third edition of this book, but I don't think the fourth could be much better.

First of all, this title shouldn't be included in the "Graduate Texts in Mathematics" series becausethe material it covers is covered in introductory undergraduate courses.Second, eventhough the author made a great effort to include as much topicsas he could, the treatment of most of them is highly old-fashioned. I mean,he pays no attention to the most recent and elegant refinements of thebasic theory, so the student is not immediately able to understand the realimportant ideas behind the subject. For example, nowadays the proof of theCauchy integral formula is presented as a more ar less easy corollary ofthe general Stokes theorem. The Cauchy integral theorem is also obtainedeasily following the same fashion. Incredibly, the author explores thisline in one appendix, but not well done, and apparently he doesn't realizethat there is the key idea.

Also, keeping in mind that holomorphicfunctions are harmonic, most of the important results for holomorphicfunctions should follow at once from the corresponding ones for harmonicfunctions, but this old-fashioned texts don't take this remarkableimportant feature of complex analysis into account, making the treatmentinnecessarily complicated and leading the student to misunderstand bothcomplex and harmonic analysis. Eventhough the book includes a whole chapteron harmonic functions, the author doesn't use their power as heshould.

I'm afraid there are few famous introductory texts that I wouldsuggest for first-timers. The best of them is Markushevitch, unfortunatelyout of print.

There is also another serious drawback: The author pays noattention at all to boundary value problems and therefore to theCauchy-type integral, maybe the most important tool of complex analysis.The Hilbert transform is also not present.

If you have the opportunitytake a look at Muskhelishvili's "Singular Integral Equations" andGakhov's "Boundary Value Problems" and then you will understandmy point.

Lang's book could be used as a companion text and as areference for introductory courses. It's got some interestigexcercises.

Its contents are: Complex Nubers and Functions; Power Series;Cauchy's Theorem, First Part; Winding Numbers and Cauchy's Theorem;Applications of Cauchy's Integral Formula; Calculus of Residues; ConformalMappings; Harmonic Functions; Schwartz Reflection; The Riemann MappingTheorem; Analytic Continuation Along Curves; Applications of the MaximumPrinciple and jensen's Formula; Entire and Meromorphic Functions; EllipticFuctions; The Gamma and Zeta Functions; The Prime number Theorem;Appendices.

Please take a look to the rest of my reviews (just click onmy name above).

Reviews (1)

The Great American Differential Geometry Book
Michael Spivak begins these five volumes stating his modest aim to write the "Great American Differential Geometry book." He surely has.Instead of listing the numerous subjects Spivak treats clearly and beautifully in these volumes, I'd like to call out the delightful travelogue style in which they are written, using history, anecdotes, and opinion to explain, illuminate, and, when possible, motivate the gleaming modern edifice. Spivak's opinions are sprinkled lightly here and there like easter eggs. How could you not love a math book that uses the subtitle "The Debauch of Indices," or dismisses Eric Temple Bell's history as "supercilious remarks of questionable taste"? Also, don't miss the annotated bibliography in volume 5. The fact that legions of professionals refer to these books in their original *typewritten* format [1st & 2d editions] is a further testament to their quality. The third edition is typeset using TeX and, though beautiful, still manages to retain a little of the quirky typewritten appearance. One quibble: I was disappointed to see that this edition did not use Richard Bassein's bizarre artwork [think 70s psychedelic] for the covers; I admit that this stuff weirded me out originally, but have grown to love it -- where else could I see fuzzy trolls in crowns made from Enneper's minimal surface?

Reviews (1)

Reviews (24)

The best rigorous introduction to general topology!
I have owned the 1975's first edition (red cover) of this book which I am currently studying again to pass a Ph.D. qualifying exam on topology. From the many topology texts that I have come across over the years, this one easily stands out as the best rigorous introduction to point set topology for a beginning graduate student. It covers all the standard material for a first course in general topology beginning with a chapter on set theory, and now in the second edition includes a rather extensive treatment of the elemantary algebraic topology. The style of writing is student-friendly, the topics are nicely motivated, (counter-)examples are given where they are needed, many diagrams provided, the chapter exercisesrelevant with the correct degree of difficulty, and there are virtually no typos.

The 2nd edition fine tunes the exposition throughout, including a better paragraph formatting of the material and also greatly expands on the treatment of algebraic topology, making up for 14 total chapters (as opposed to 8 in the first edition). A notable minor issue in the first edition was the consistent usage of the pronoun "he" in the discussions for addressing all the possible readers of the book. (This fortunately has been modified in the 2000's edition.) On another note, I wish there were some hints & answers provided at the back of the book to some of the harder problems, so as to make this text more helpful for those of us who use it for self-study.

One of the two spotlight reviewers has correctly mentioned that Munkres does not cover differential topology here. I speculate this is perhaps because Munkres has already a separate monograph on differential topology. It is also necessary to get a handle on some fair amount of algebraic topology first, for a full-fledged coverage of the differential treatment. Regardless, one great reference for a rigorous and worthwhile excursion into differnetial topology (covering also Morse Theory) is the excellent monograph by Morris W. Hirsch, which is available on the Springer-Verlag GTM series.

At the end, I shall mention that one other very decent book on general topology which has unfortunately been out of print for quite some time is a treatise by "James Dugundji" (Prentice Hall, 1965). The latter would nicely complement Munkres (for example, Dugundji discusses ultrafilters and some more of the analytic directions of the subject.) It's a real pity that the Dover publications for example, has not yet published Dugundji in the form of one of their paperbacks.

great!
Not much to add here... there are enough easy problems that I can get the hang of something, but also some really tough ones at the end of each problem section. The proofs and examples in the text are really good guides to doing the problems also. In some sections there are counterexamples for, say, the converse of a theorem which are always really pathological. At the beginning of each section there is some discussion on what to expect, why the stuff is important, what to do with it, etc. Even though I had a really good prof for the topology course I did this book was very helpful out of the classroom.

Excellent Topology Book
My introduction to Munkres was in an independent study of point set topology in my final semester of undergraduate work. A professor assigned me problems from the book, but my learning was largely self motivated.I found that it was an excellent book for independent study.The text was clear and readable and the exercises helped to cement the concepts that are introduced in the reading.

Later at graduate school, Munkres was also used in a topology class at the beginning graduate level.Highlights were taken from the first section (point set topology), and a large focus of the class was on the algebraic topology in the second section of the book.Sometimes I had difficulty following exactly what the professor was doing at the blackboard, but I could always understand what was going on when I consulted Munkres.

Reviews (4)

Excellent Introduction to Combinatorics
A COURSE IN COMBINATORICS covers a great breadth of topics under the label of "combinatorics," including graph theory, enumeration, and some algebra. The book is comprehensive; the instructor can pick-and-choose appropriate material from the huge array provided without detriment to understanding.

Each chapter is written in a friendly, accessible manner: plenty of interesting and instructive examples follow the clear definitions and preliminaries. To give the reader an idea of the topics presented in the book, a list of chapters follows:

1. Graphs
2. Trees
3. Colorings of graphs and Ramsey's theorem
4. Turan's theorem and extremal graphs
5. Systems of distinct representatives
6. Dilworth's theorem and extremal set theory
7. Flows in networks
8. De Bruijn sequences
9. Two (0,1,*) problems: addressing for graphs and a hash-coding scheme
10. The principle of inclusion-exclusion; inversion formulae
11. Permanents
12. The Van der Waerden conjecture
13. Elementary counting; Stirling numbers
14. Recursions and generating functions
15. Partitions
16. (0,1)-Matrices
17. Latin Squares
18. Hadamard matrices, Reed-Muller codes
19. Designs
20. Codes and designs
21. Strongly regular graphs and partial geometries
22. Orthogonal Latin squares
23. Projetive and combinatorial geometries
24. Gaussian numbers and q-analogues
25. Lattices and Mobius inversion
26. Combinatorial designs and projective geometries
27. Difference sets and automorphisms
28. Difference sets and the group ring
29. Codes and symmetric designs
30. Association schemes
31. (More) algebraic techniques in graph theory
32. Graph connectivity
33. Planarity and coloring
34. Whitney duality
35. Embeddings of graphs on surfaces
36. Electrical networks and squared squares
37. Polya theory of counting
38. Baranyai's theorem

The problems in the book are generally very rich and well-written, with helpful hints from the appendix that provide motivation but do not spoil. However, the relative difficulty of the problems is not readily made appparent, so over- or underthinking of problems often occurs with misjudgments.

For the interested high-school student to the beginning graduate, this book is ideal for the study of combinatorics. Truly a nice read that connects many areas of mathematics and combines them into a thing of true beauty.

A nice tour of combinatorics
The first word that comes to my mind when I think of this text is "encyclopedic". It contains around 40 chapters, hitting most of the high points of combinatorics that a graduate student should see. The exposition is generally good with nice examples. The one thing that I fault it for is the number of statements that the authors claim are "obvious". In a way, this is good, because it makes you pay attention and understand the material, but sometimes the statement isn't obvious until you've thought about it for an hour and written out a lengthy proof. At that point, it does become completely obvious and you can't believe that you ever thought it wasn't, so I can understand why van Lint and Wilson fell into the trap so often. (In fact, I've heard that Wilson even stumbles over some of those points in lectures.) This is a great book to have on your shelf if you need somewhere to look up combinatorial ideas.

A gentle introduction to combinatorics
This book was the text for a graduate-level course I took.The presentation is very laid-back, much like the lecturing style of one of the authors (Wilson), and so it was quite readable (unlike many other mathbooks which you have to stop every few pages and pick apart everythingbefore it sinks in).

Combinatorics is a relatively recent development inmathematics, one which is generally easy to explain, but with manydifficult open questions.Van Lint and Wilson do an excellent jobexplaining, but there are a few places where the reader needs to know somebackground to place the particular problem in the appropriate mathematicalcontext.Understandably, if the authors were to include all themathematical machinery needed, the book would be huge!Instead, they havechosen to describe as many facets of the field as possible, and thereforehave written a broad, well-balanced book which approaches the topic in anon-threatening way.

My one criticism, then, is that there is a lack ofdepth in several areas of the book, with further discussion of advancedtopics or open problems.But even so, I can appreciate the omission forthe sake of accessibility.

Reviews (6)

One invaluable pearl
This book is an invaluable pearl in mathematics.
Both content and appearance of the book are excellent.

A breath of pure air
I stumbled across this book and am amazed that I had not heard about it before. Since buying it, I have kept it by my bedside and have now read the whole book four or five times, picking up more of the subtleties at each reading.

The proofs are almost all magnificent (although I wonder how Buffon and his needles got in there) and even the well-known and time-honoured ones have a new twist or new extension.

The level of mathematics required to follow the proofs is reasonably low (high-school 'A' levels in the British system, no idea about other countries) although the book gives a deeper explanation in some areas (e.g. trans-finite arithmetic) than in others (e.g. number theory). I wonder if this unevenness reflects the interests of the authors.

But these are tiny nit-pickings. This is a wonderful and inspiring book and reading it should be made compulsory by the government in all high-school mathematics classes.

A fitting tribute to the great Paul Erdos
Paul Erdos once remarked that you need not believe in God, but you certainly have to believe in the book in which God maintains the "perfect" mathematical proofs. Martin Aigner and Gunter Ziegler have certainly done a great job with this book, a fitting tribute to the great Erdos himself.

I had purchased a copy of the 1st edition of this book and was plesantly surprised that the authors had come up with a 2nd edition, with a few more "perfect" proofs.

My personal favorites are "The Shannon capacity of a graph". where the Lovasz theta number would eventually lead to semidefinite programming, Erdos' probabilistic method where probability makes counting sometimes easy, computing the number of trees in a graph, how many guards it takes to guard a museum, and the section on Turan's theorem.

Reviews (5)

The New Goldstein
If you have no exposure to Classical Mechanics or if you starting new then DON'T BUY THIS BOOK. This book is of advance nature. Books like Arya, Hauser or Corben are simpler.

The book starts with the basics, wraps up Newtonian Mechanics in the first chapter. The next five chapters deal in detail with Lagrangian and Hamiltonian formulations and their applications. They discuss some difficult topics like "Tangent Bundle" "Tangent Spaces" and "lie groups". "Noether's theorem" is also included which I have not seen in many Classical Mechanics books.There are many worked out examples which help the reader understand the subject.The explanations are quite lucid.There are plently of excercises to do but no answers to problems!

A Contemporary Textbook on Mechanics
This book offers all the standard text used in undergraduate mechanics courses plus a number of more contemporary topics such as Lie derivatives, manifolds and much on nonlinear dynamical systems, all in a language appropriate for a Physics book. I consider it to be the modern equivalent of classics like the books by Goldstein or Marion. The material of this book should be the new standard for modern Classical Mechanics courses.

An excellent book on a fundamental subject
Classical mechanics often falls by the wayside in a modern physics curriculum. However, there are times when an understanding of subtle issues in this field are simply necessary for progress in current researchdirections. At times like these, one is all-too-often forced to turn toolder texts such as Goldstein or directly to the literature of a field withwhich one is rarely intimately familiar. It is therefore a great pleasureto find a text such as Jose and Saletan's, a highly modern, extremelycomplete and very readable textbook on mechanics at an advanced level.

The book covers all of the standard topics of a graduate mechanicscourse (Lagrangian and Hamiltonian dynamics, rigid bodies, etc.) as well asmore modern topics such as chaotic dynamics. All these subjects are treatedin great detail and both in very physical and very formal languages. Mostimportantly, all of these discussions (including the formal ones!) arepacked with completely worked examples which allow one to begin to usethese techniques without attempting to decipher formal proofs.

Reviews (6)

how many stars??
I read this page to see how Milnor's book could average only 4 stars.I found two unfair, but intelligent, 2/3 star reviews apparently by physicists, miffed that a masterful 50 year old book on the foundations of the topic, does not provide a history of their invention, and a survey of their current use in physics.If a book serves its own and its author's purpose wonderfully well, but does not serve yours, that is your fault, not the author's.As a hint, it is in the Annals of Math series, not the Handbook for Physicists series.Please read the chapter on "obstructions", p. 139.There Milnor succintly provides exactly the interpretation requested, and a reference to the classic text by Steenrod.

The most important construction on a smooth manifold is its tangent bundle, and the basic question is whether smooth never zero vector fields exist.The subject begins with the theorem of Poincare & Hopf:a never zero vector field exists if and only if the topological euler characteristic of the underlying manifold is zero.For a polyhedron, this euler characteristic is the number V-E+F = vertices -edges + faces.Thus the most basic characteristic class is the euler class.Briefly, the others measure existence of sequences of independent vector fields.In 1957 their existence, construction and properties were clouded, and Milnor cleared this away once for all in these notes, published by demand and gratefully received by [almost] everyone.This is a great book, and a 2 star review only serves to rate ones own qualifications to appreciate it.I.e. these reviewers are rating not the book but its suitability for their own narrow interests.For another short introduction try the chapter in the book by Bott & Tu.

A mixed bag
Coming from a physics point of view, I did not find this exposition the most useful. It is rather formal and the ordering isn't that natural for a physicist. There is very little effort expended on examples to build intuition.
Nevertheless, it may be the only game in town. I haven't looked into what other books are available. I do know that the ones I have seen intended for physicists are pretty thin.

Great!
The point to bemade here is that M&S and books comparable to it ( I can think of thoseby Morita, off hand) are written in a style amenable to mathematicians. The purely formal, albeit axiomatic, approach survives as it appeals more to purists than to physicists.There's really nothing lacking in MS:although dated, it's very readable, requiring only a minimum of prerequisites,and this is what makes it attractive, with an appeal to a wide spectrum of audiences. Physical applicationsare bountiful, and they can be sought elsewhere in the literature.

Isbn: 0821834363
Sales Rank: 500066
Subjects:  1. General    2. Knot theory    3. Link theory    4. Mathematics   

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 (5)

This book is an affront to mathematics!
There is, no doubt, a number of reviewers who will find
this book beautiful for its comprehensive coverage of
the subject.Unfortunately, this book has now been almost
universaly adopted as a default introductory text.As such,
it fails miserably.
The author assumes much more than what is commonly refered
to as a "degree of mathematical maturity".He almost universally
fails to provide references to the facts previously proven when
using them in arguments.The presentation most resembles a
conversation someone would have in a bar after patronizing the
place for some time.Before turning the page of the book, it
would be higly recommended to have committed to memory the entire
contents of the book up to that page.Rigor is present nowhere
in the book.
In truth, the book is comprehensive.But it preaches to the
converted.Algebraic topology is a beautiful subject that will
be hated by the present generation of mathematicians because
they will study it from Hatcher's book.

Very good book, but don't buy it!
This book is avaliable free to download from Allen Hatcher's webpage. You will also find other books he has written.

http://www.math.cornell.edu/~hatcher/

It's worth your money!
This book is not just for topologists!If you're like me, then you've spent countless nights sans Hatcher's book trying to figure out the fundamental group of a beer can.Look no further, the answers are here!

Reviews (7)

Lucid and elegant, but not for beginners
This tiny textbook is well organized with an incredible amount of information. If you manage to read this, you will have much machinery of algebraic topology at hand. But, this book is not for you if you know practically nothing about the subject (hence four stars). I believe this work should be understood to have compiled "what topologists should know about algebraic topology" in a minimum number of pages.

An Outstanding Achievement.
As a matter of fact all the materials written by J. P. May are precise, concise and useful. He is not of kind of those people who write 1000 pages and reach at obvious matters. This book is really a good introduction to the modern aspects of algebraic topology. It has less than 250 pages. I liked the treatment very much and appreciate it for teaching me a lot of mathematics. I dare to say that if someone else wants to write a book including all materials treated in this book, then the book would consist of at least 1000 pages. There is more to this book than just classical algebraic topology.

A Unique and Necessary Book
Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology. It is really necessary to draw pictures of tori, see the holes, and then write down the chain complexes that compute them. Likewise, one should bang on the Serre Spectral Sequence with some concrete examples to learn the incredible computational powers of Algebraic Topology. There are many excellent and elementary introductions to Algebraic Topology of this type (I like Bott & Tu because of its quick introduction of spectral sequences and use of differential forms to bypass much homological algebra that is not instructive to the novice).

However, as Willard points out, mathematics is learned by successive approximation to the truth. As you becomes more mathematically sophisticated, you should relearn algebraic topology to understand it the way that working mathematicians do. Peter May's book is the only text that I know of that concisely presents the core concepts algebraic topology from a sophisticated abstract point of view. To make it even better, it is beautifully written and the pedagogy is excellent, as Peter May has been teaching and refining this course for decades. Every line has obviously been thought about carefully for correctness and clarity.

As an example, ones first exposure to singular homology should be concrete approach using singular chains, but this ultimately doesn't explain why many of the artificial-looking definitions of singular homology are the natural choices. In addition, this decidedly old-fashioned approach is hard to generalize to other combinatorial constructions.

Reviews (7)

A review from a graduate student
If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following:

1. Complex Analysis
2. Differential Geometry and calculus on manifolds
3. Homology-Cohomology Theory
4. Undergraduate Algebraic Geometry

Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.

However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.

So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.

algebraic geometry: the real stuff
The book is beautifully written and easy to read, with emphasis on geometric picture instead of abstract nonsense. By far the best introduction to algebraic geometry for string theorists.

Reviews (1)

Reviews (1)

Isbn: 0521809061
Sales Rank: 582699
Subjects:  1. Algebra - Linear    2. Algebraic Geometry    3. Geometry - Algebraic    4. Geometry - General    5. Invariants    6. Mathematics    7. Moduli theory    8. Number Theory    9. Science/Mathematics    10. Theory Of Numbers    11. Mathematics / Geometry / General    12. Topology