Reviews (8)

Old fashioned, unpretentious and solid.
It's kind of like having a friend explain it to you.Also dips into fractions, continued fractions, irrational numbers and other non-integer stuff. I find it interesting how the style of most older math texts differs from most current writing in that the older texts are much friendlier.Also, older books usually lack the annoying drizzle of typos and blunders which mars most recently published "higher" math books.Some of this may be due to correction opportunities afforded by various editions and updates, but I suspect a higher standard of care in the original writing and editing explains a lot of it.The book is sadly overpriced, of course, but that's life.

One of the greatest
First of all, let me say this about the one star review. Do not let yourself be infuenced by lesser mathematicians. Idiots in my opinion. To give this book one star, you must posses some special kind of mediocracy. Keep your stupidity to yourself Lucas.

No one writes like this anymore. Mathematicians like Hardy have passed. The subject has ballooned, and now you have to specialize within Number Theory. There are fewer and fewer that can posses knowledge of the entire subject of Number Theory. Remember what Harold M. Edwards said. You have to read the classics, and beware of secondary sources. Authors give their own spin on ideas. And who is to say they have a greater or lesser understanding of the subject. Furthermore, who can determine how well can they express themselves. How many mathematicians our days bother to study grammar and literature? The best example is Gauss' Disquisitiones Arithmeticae. Would you rather read a book written by Gauss himself, the man that established the subject? Or by some one who learned what some one learned what some one learned over a period of 200 years? Also know what Axler, author of Linear Algebra Done Right, said about reading mathematics books. For a mathematics book, if you spend less than half an hour per page you are going too fast. The last thing i will say is again attributed to Edwards. In his book on Advanced Calculus he encourages the reader to jump chapters. A book does not have to, and sometimes it should not, be read in order. It may take some practice to see how you need to jump around, but you will find that you can maximize your reading by doing so.

There are several point in which this book excels. First, in the writing style. Second, in how many ideas it introduces. Or how good an understanding the reader obtains of Number Theory. It is invaluable to have the big picture. Third, the author has in mind the future material the reader will encounter. He knows you will go beyond this book, and prepares you for what is to come. You do not enter higher courses blind.

The writting style is representative of that of Wiles and Loiville. It will show you how your mathematical writting should be. It takes a lot of practice to learn mathematical formalism and how to write proofs. This is the book to learn from. The author is not afraid to connect the ideas you are learning to other advanced ideas and to mathematical history, unlike present day authors. If you plan to be a mathematician, you must know its history. The writting is in a mathematical sense superfluos. It does not assume you are a genius, but strikes balance between what you should know and what you should be told.

The book is successful in providing you with the big picture, and how ideas you are learning reflect one ideas you will learn or have already learned. Having a big picture of the subject, which he describes in the second chapter, lets you know what you are learning now and puts the entire material in context. Gives you great perspective of the subject. Because a great deal of branches of number theory are discussed, you are not only better equiped to choose which branch might interest you, but it eases the transition to more advanced courses, such as Analytical Number Theory.

The author from the start discusses unanswered questions in Number Theory. I know alot of professors which think that the student should not be exposed to questions that surpass his mathematical knowledge. They are the weak mathematicians. Mathematics is about exploring and breaking limits. You should know what is beyond your reach, and the reach of every one else. The questions that still stand might be answered by some one that was intrigued by the challenge of answering them when they are helpless to do so. Fermat's Last Thorem is such an example. The guy learned it at the age of 10.

The last thing i will say about the book is this. Number theory has one scope. Namely, prime numbers. This book make it clear that the purpose of number theory is to determine the properties of numbers. It discusses the limitations of mathematics in attaining answers to Riemann Hypothesis, Fundamental theorem, trancedental and irrational and algebraic numbers, and so on. Thebook is, in my opinion, an expansion of the section on unanswered questions. And in doing so many more questions are asked and analyzed. There are prime numbers, and nothing else.

Reviews (2)

Outstanding Title
I guess that, besides the classic references from Pazy and Jerome Goldstein, this title outstands in its category. I've heard several words of criticism over Pazy's work due to the lack of applications and several laments over the fact that Goldstein's book is out of print. This book introduces the semigroup theory like no other: it doesn't forget the historic and philosophical aspect of subject and it's full filled with applications and deep beauty. Congrats to the authors and also to the customers that decide to buy this title, may it be PDE or Dinamic Systems, you'll be home free...

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 (2)

A Course in Functional Analysis ... the title is correct!
I learned functional analysis by studying this book. I did this under the direction of a master teacher, John Erdman, who taught via a modified Moore Method. I found this very inspirational and challenging. BEFORE I took thecourse, I did not enjoy browsing the book, BUT I learned that the book,upon combination with the right amount of focus and effort, did aremarkable job of bringing functional analysis alive ... of transmiting thereal essence to young, "sprouting" mathematicians. There is alsoan informality that brings a freshness to the book ... and this in asubject that could easily be studied without encountering this importantingredient in a mathematician's training.

This book has as it's highpoint and goal the spectral theorem for normal operators. I add thisbecause no one book can be all encompassing. If this and the spectraltheorem goal are kept in mind, the omissions and emphasis found in the bookwill be found to be completely natural.

This book should be in thelibrary of anyone teaching functional analysis or who wants a workingmathematician's masterfully developed course on functional analysis (withan eye to the spectral theorem for normal operators).

Reviews (21)

Wedderburn, Waring and Hamilton
Not necessarily in that order.President McCosh of Princeton
waxes eloquent in his Scottish Philosophy book somewhere on
the internet, re: Dugald Stewart, Kant and Hamilton.

Hamilton is a strong vice, but clearly represented in Herstein.

Good Introduction, useful for self study
I am an engineer by training and a sales man by profession, with a a strong liking for mathematics.
I found this book to be an very readable introduction to a subject (abstract algebra), I had never been exposed to during my engineering math - other than matirx theory, which was obviously taught extensively.
The proofs are generally easy to understand, but certainly not trivial.
A pleasure to read

Reviews (24)

Exceptionally pretty complex analysis book
This is a very exciting introduction to complex analysis. Its most striking feature is the many excellent illustrations; pictures are used to explain things whenever possible. Needham is always eager to explain, and also to show meaning. Thus Möbius transformations are not just charming quirks; instead Needham gives a self-contained introduction to non-Euclidean geometry to show them in action. And when one needs to understand analytic functions as flows, then Needham gives a self-contained introduction to the ideas of vector analysis. Because the book always spills over on other topics like this, I keep my copy within reach at all times, as a treasure mine of beautiful, visual explanations of topics even outside of complex analysis proper. This wide scope works very well most of the time but it should be said that there are probably too many minor side topics than are appropriate in a first textbook. Sometimes Needham seems to include results not because they fit in or add something important, but because he has thought of such a pretty proof. But we quickly forgive him, for the book is so extremely loveable, and it is still by far my first choice as a first textbook of complex analysis.

essential
It is possible to memorize definitions, master proofs and work endless exercises and still feel that you don't understand what's going on.This is especially true in complex analysis.This book emphasizes the visual/geometric aspect of analytic functions at the expense of some loss of rigor.These insights are priceless.

Needham employs a wider range of mathematical tools than other books aimed at the upper level undergraduate, e.g., Palka and Brown/Churchill.This would include simple group theory, linear algebra, vector calculus and obviously geometry.Often this works very well at the expense of some digression. At other times, the more traditional algebraic approach is better.

This book is unique and fills an important need.

Reviews (9)

overrated, but still good
This book is a good reference, because it contains a lot more material than is contained in most courses, but I don't think I'd want to use it for an intro to linear algebra. It's got stuff that other books don't have, like Hilbert spaces & some analysis stuff in an appendix, tensor products, multilinear forms... It's good as a reference or supplement, but not as a main text, IMO. For an intro, I liked Axler's Linear Algebra Done Right or the Hoffman/Kunze book.

Linear algebra for mathematicians
I've just been looking on Amazon to see how some of my favorite old math texts are doing. I used this one about twenty years ago as a supplementary reference in a graduate course, and I still have my copy.

Everybody with some mathematical background knows the name of Paul Richard Halmos. I saw him speak at Kent State University while I was an undergraduate there (some twenty-odd years ago); to this day I remember the sheer elegance of his presentation and even recall some of the specific points on which, like a magician, he drew gasps and applause from his audience of mathematicians and math students.

This book displays the same elegance. If you're looking for a book that provides an exposition of linear algebra the way mathematicians think of it, this is it.

This very fact will probably be a stumbling block for some readers. The difficulty is that, in order to appreciate what Halmos is up to here, you have to have _enough_ practice in mathematical thinking to grasp that linear algebra isn't the same thing as matrix algebra.

In your introductory linear algebra course, linear transformations were probably simply identified with matrices. But really (i.e., mathematically), a linear transformation is a special sort of mathematical object, one that can be _represented_ by a matrix (actually by a lot of different matrices) once a coordinate system has been introduced, but one that 'lives' in the spaces with which abstract algebra deals, independently of any choice of coordinates.

In short, don't expect numbers and calculations here. This book is about abstract algebraic structure, not about matrix computations.

If that's not what you're looking for, you'll probably be disappointed in this book. If that _is_ what you want, you may still find this book hard going, but the rewards will be worth the effort.

The best abstract linear algebra book out there
This book is the best if you are looking for an abstract approach to linear algebra. It provides elegant proofs to theorems that usually seem long-winded and awkward (like the cauchy-schwarz inequality). Sometimes in your lectures you may get to the point thinking "can't this be proven more elegant?" and you simply open halmos and it is there.

Note that this book does not deal alot with matrices, everything of the theory is there, but you might miss illustrations and applications. In this case I recommend to back it up with Gilbert Strangs Linear Algebra and its Applications, which has an intuitive, matrice-oriented approach.

Isbn: 0140227873
Sales Rank: 2642004