Wow! What a great Table of Contents. It has all the stuff I've been wanting to learn about. So I bought the book in spite of seeing only one review of it. After one day, I'm now only at page 26, but I already have read enough to make some comments about it.

The main point about this book is that it is, as the author specifically states, LECTURE NOTES, not, I repeat, not a textbook. What are the implications of this (outside of a somewhat more chatty style than a textbook)? ["chatty" isn't quite what I mean; "smooth" might be a better word'] There are two which are noticable to me. 1) A lot of math knowledge is taken for granted. 2) It has a somewhat sloppy style to it.



Regarding point one, make sure you have a lot of math under your belt before picking up this book. By page 18 the author uses these terms without defining them: Differentiable Manifold, semigroup, Riemannian Metric, Topological Space, Hilbert Space, the "" notation, vector space, and Boolean Algebra. Fortunately for me, I have a fairly extensive math education, and self-studied Functional Analysis, so I wasn't thrown for a loop; but for many others -- brace yourselves!



Regarding point two, Here are two examples:

1) Here is a quote: "The collection of all open sets in any metric space is called the topology associated with the space." Sounds like a definition to me! Fortunately the author gives a (sloppy) definition a few lines later. By the way, the only thing the reader learns about what an 'open set' is, is that it contains none of its boundary points. All the topology books I have read define open sets to be those in the topology. This is another point of confusion for the reader. In fact, points of confusion abound in that portion of the book.

2) On page, 17, trying somewhat haphazardly to explain the concept of a neighborhood, the author defines N as "N := {N(x) | x is an element of X}" This is already a little disconcerting: x is already understood to be an element of X. So he is saying that N is defined as N(x) (which he defines to be a collection of subsets of X). This is all he has to say on the matter until, on page 26, he writes "each N, an element of N(x)". Now N isn't both N(x) and an element of N(x). This is a point which the author does not clear up. He then starts using N all over the place, yet the reader isn't sure of what he's refering to.



A couple of other things:

-When he defines terms, they is not highlighted, and are embedded in a sentence, making it difficult to find them later.

- The index is pitifully small. Typical for English texts, I know; but this *is* the 3rd millinium!



On the other hand, I have good things to say about the book, too.

I like his style of writing. If it were just more precise, it would be fine for me. I like it better than the normal higher math texts, which tend to be too laconic for me. Notice that I make a distinction between the somewhat chatty style, which I like, and the sloppiness, which is confusing. One can be chatty, yet clear. So far, the undefined math terms which I listed above were not central to the text; and one would not miss much by just reading past them. The author includes many 'comments' sections throughout the book. These are wonderful so far. They are full of comments and examples which really clear up a lot of points. His examples are very good, too, although he is very terse in stating them. The paperback is nice looking. The paper, font, etc. make for easy reading (except for the sub/super-script font, which is too small for me).



To wrap this review up, I had already pretty much learned the stuff covered in the book so far, but judging from what I have read, I will be able to learn a lot from the rest of it; and, unlike some other math books I have studied, the experience won't be too painful.

p.s. See other reviews of it on the UK Amazon site.