Foundations of Mathematical Analysis

I am currently reading this book so I can learn calculus the "right" way. My undergrad courses in advance calculus and complex variables (notice I say variables and not analysis) were written for engineers and science majors that needed to know math, but not at this level. I find the sequence of topics (sets and functions, real numbers, sequences, ...) extremely helpful for understanding the material. One topic leads to another in a very logical and progressive manner. The problems range from the very easy "one liners" to more complex problems. The book contains hints to solving them at the end of the book. Very nice for self-teaching.

The pro's : Very good, everything is explained in a clear way, starting from the beginning, no gaps left in the proofs, and the material is abstract enough to motivate math lovers...In fact every math undergraduate and graduate should have this book as a reference, this cannot be a problem when you see the price. The con's : Dover always has cheap price editions. While there is definitely a market for this, let' s face it : these editions have some disadvantages : While the contents of this book are very well suited not only as a reference but also to learn the material, the dense layout is not so comfortable to learn from. In that sense, the authors deserve a better edition... Maybe a question to the Dover guys : is it possible to bring your excellent science books in two editions : the existing cheap editions, and another more comfortable edition : bigger size, more whitespace on each page, ....

A very good introduction to real analysis, with all the appropriate theorems and proofs presented in a well ordered and understandable fashion. I use it regularly in my own teaching and research.

Everybody seems to think that Rudin's Principles of Mathematical Analysis is the best for whatever reason, & I agree that it's good for reference after being exposed to the material. Pfaffenberger doesn't construct the real numbers using Dedekind cuts, he makes a list of 13 axioms that basically say that the reals is the only complete ordered field. I think I liked this approach better than Rudin's (or Hardy's) more abstract approach. He also spends much more time developing metric spaces (including the Baire Category Theorem & nowhere dense sets, etc, which Rudin omits except for a couple exercises) and the Riemann-Stieltjes Integral. Then there's a short chapter where transcendental functions exp, sin & cos are defined which I think Rudin skips, and then introduction to inner-product spaces. Fourier series is introduced in the chapter on general inner-iroduct spaces (with a first look at Banach Spaces as an aside) rather than the chapter on sequences & series of functions. I also liked this better than Rudin's text or other calculus texts. Rudin includes a whole chapter on functions of several variables, but Pfaffenberger doesn't have anything on them. Instead, there's a chapter on normed linear spaces and the Riesz Representation Theorem, and then similar to Rudin, a chapter on the Lesbesgue Integral. This book has many more problems than Rudin's or Apostol's, and in general they are a bit easier. Of course every section has its difficult ones but the first ones are almost always just "verify that blah blah is true". The last ones are about on the same level as the ones in Rudin's book.