Customer Reviews

Beginning Functional Analysis

5 star:		(2)
4 star:		(0)
3 star:		(1)
2 star:		(2)
1 star:		(1)

 

 

This book is fantastic! It is an extremely readable account of the basics of the subject. I thought the Chapter on Measure Theory and Lebesgue integration were particularly well organized. Every definition was well motivated and the theorems were arranged in a very natural progression.

One thing I especially enjoyed about this book is that most of the proofs are done only for special cases of theorems, without loss of generality. For example, the Arzela-Ascoli theorem is proved for the function space C([a,b],R) (R = real numbers), but then Saxe points out what makes the proof 'tick' so that the reader may easily modify it to a more general setting (she always states the more precise versions of such theorems as well). This is great because it helps one's intuition without getting short-changed.

Finally, the book has a great wealth of historical notes and biographies which are rich in mathematical content (e.g., Saxe explains that Frechet was the first person to define a metric space even though he called it 'une class E'; Hausdorff gave it its modern name in 1914). The reader can in this way appreciate how the subject slowly developed into its present form.

This book is a jewel! I myself am not the biggest fan of functional analysis, but this book made me really appreciate the subject.

I do not agree with the other readers that say this book moves too quickly or is too difficult--your first couple encounters with the UTM series often always seem this way. However, I don't think this is the best introductory book on the subject material for a number of reasons. The other reviewers make clear that less experienced students do not find this book easy--I am pointing out that a more experienced student finds this book boring.

I feel that functional analysis, like linear alegbra, is one subject in mathematics that is often not sufficiently motivated, and as a result comes across as both difficult and boring. While I think this book takes some of the difficulty out of things through its straightforward and clear explanations of various concepts, and through its relatively easy exercises which reinforce the material, I find this book to be exceptionally boring, especially in the initial few chapters. The book is too concise for a UTM text; it's hard for students to understand why to get through the earlier material unless you give more interesting examples and talk a little bit more about where you're going and why anyone would want to learn this stuff.

I would recommend the book by Kreyszig as a supplement to anyone stuck reading this one; it doesn't cover the same material (it tends to be a little less advanced, omitting discussion of measures), and it's a very different sort of book, but especially in the earlier few chapters it will provide a much needed energy, more interesting examples.

These past two semesters I've been enrolled in a graduate-level analysis course. The book we used, by Folland (see my review) was not a good book, in my opinion. After reading the first several chapters of Folland, I just did not understand what the point was to Lebesgue integration, or why we had to develop all this machinery that goes along with it.

Sometime during the semester, I got hold of this book, by Saxe, and started reading the chapters on Lebesgue integration. After doing that, I began to develop an understanding of what it was, how it was used, and why it was necessary to cover all these theorems. The book gave me perspective on the subject; (and hence motivation) something which Folland did not do.

Saxe's book isn't without it's faults; I had some issues with her proof of the Baire Category theorem (in this case, I actually found Follands proof much more believable) and she got the year of Hermann Minkowski's death wrong. Other than that, I could not find any problems with the book.

In summary, this book fills a much needed void in the literature: a readable book which introduces the student to functional analysis beyond the undergraduate "advanced calculus level." If you are in a graduate-level real analysis course and haven't a clue what a sigma algebra is or why you should care (but would like to), then buy this book.

I wouldn't call this an undergrad text. It's more advanced and moves faster than Rynne & Youngson. In an effort to cover a lot without much prerequisites, it moves too fast, introducing a lot of basic ideas in an offhand way. I found the explanations generally not as clear as in related books. No examples, few hints, no answers.

A lot of bla bla in this book. This book is a collection of notes without a guideline in mind. The more interesting subjects of functional analysis are only superficially treated. Proofs are often quite shortly illustrated and in a proof I found a gigantic error. Not worth to buy this book.

I'm sorry I had to read this book, I'm sorry anyone else had to read this book, and I hope that no one else has to suffer through this book. Pedagogically speaking, it is un-sound to state, in the first chapter and in the first several pages, a theorem and, rather than set a tone and rhytmn for the book, good 'ole Saxes leaves these theorems to the reader as exercises. Now, Saxe is not Lang and the Springer book series, namely, undergraduate texts, is just that, for undergraduates, for young students, mathematically speaking, and hence such an adverse and lazy tone should not be set in such a book. Furthermore, her direction, her flow, her in-ability to illuminate and bring forth the beauty of Functional Analysis is depressing (see Ascoli's Theorem and, while your seeing things, maybe Saxes should have seen an editor). For those of you who have had some Analysis, are familiar with basic point-set topology, and have a general idea of what it means for a mapping between to spaces to be linear, then the book you should read is that of Erwin Kreyszig. This book provides you with all the necessary tools as well as motivated and complementary exercises (some solutions are provided). Steer far away from Saxe's book. As an aside, it is obvious that no book, and hence no author, is perfect or capable of producing a 'perfect' book. The problem that arises in the case of the current book being reviewed is that considerable knowledge and or passion for a given subject does not necessarily imply that you are capable of re-casting this knowledge in the form of a book. Maybe, just maybe, if there is a future reprint, hopefully, Saxe will be able to re-work the book and create something hjigher in quality. Until that time, go else-where if you are interested inl earning Functional Analyiss.