Introduction to Analysis (3rd Edition)

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Summary: The worst book I have never seen before
Review: I have studied this book for a semester. My professor chose this book because he thinks it easy for us to understand analysis. He professed that every student must learn Math step by step and whether the books are good or bad, any student could not understand concepts on Euclidean n space or metric space unless the concepts on R, the set of real numbers, are comprehended first.

However, this book is a counterexample. Chapter 1 to 7 discuss theorems on R including concepts of sequence, continuity, uniform contunuity, integral, convergence, series.... There is nothing about metric space or vector space until chapter 8. How could a book of analysis not mention these spaces in half of its content? I cite my professor's point of view. any student could not understand concepts on Euclidean n space or metric space unless the concepts on R, the set of real numbers, are comprehended first. It is a specious argument. Whether a student can understand or not depends on that the book chosen is easy for the student or not . It does not depend on what is learn first or second. Obviously, this book is difficult to read and understand not only for me but also for the reader who ever read the book, Elementary classical analysis 2nd edition by Jerrold E. Marsden.

I here give an example. In page 191 is theorem 7.15, Weierstrass M test. The proof uses uniform Cauchy criterion of series of function but it is not mentioned in the previous passage. Looking for it in the preceding pages, uniform Cauchy criterion of sequence of function is found in page 187 which is far away from the page of Weierstrass M test. It means that the reader can't connect the concepts used in the proof of Weierstrass M test. The book by Marsden does perfectly in this point. In page 244 in Marsden's book, Weierstrass M test follows the uniform Cauchy criterion of sequence of function immediately. Uniform Cauchy criterion of series of function is also mentioned between them. The way that Marsden wrote the book makes me connect concepts and feel easy to comprehend.

Marsden's book explains why Weierstrass M test is introduced in the passage preceding to the passage of Weierstrass M test. It also gives an intuitive interpretation to let the reader have insight into the theorem and its proof in a natural point of view, not merely a syntatic point of view like Wade's book. It applies to every theorem in Marsden's book. I could not understand the proof of Weierstrass M test in Wade's book after I read it 10 times, but I comprehended it immediately and derived the proof by myself after I read Marsden's book.

I can't comprehend Math only in syntax. I comprehend Math intuitively by reason. And I think every Math learner does in this way. A proof like the proof of Weierstrass M test can be comprehended as an a priori analytic judgement or an a priori synthetic judgement. It is said analytic because it uses Uniform Cauchy criterion. It is said synthetic because the reader can think of some examples of the theroem which are consistent with the phenomena in the real world. Thus the concepts and the theorems have objective references which can be found in our experience of phenomena. A Math learner can't merely check the validity of the process of a proof syntatically and then says he comprehends the theorem.

Rating:
Summary: This book needs more revision
Review: I have studied this book in a two semester advanced calculus basis and the reason my teacher chose the book was because it is the only book she could find that touches both singl and multivariable functions calculus. This book is said by many teachers of mine to be a very dry book and here by dry I mean that it just gives you the theory of the concept not the applications which are very important in the study of mathematics (or atleast to attract people to mathematics) I think this book is still allright at least to complete the collections of mathematics books in one's library.

Rating:
Summary: A Great Textbook for mathematics lover
Review: I have taken a two-semester course of senior level Advanced (honors) Calculus with this as text. It is a great way to learn real analysis - covering rigorous single variable analysis(first 4 chapters) in one semester - immediately followed in the subsequent semester by multivariable theory (next 4-5 chapters). The author introduces Point Set Topology not before the 5th chapter, as a tool for multivariable analysis. The 10th chapter includes an introduction to calculus on manifolds.

Rating:
Summary: Nice intro, but could use some improvements
Review: I used this book for my first analysis course. Over all, I think it does a nice job, but it could use a little tweeking. Let's start with the good...

First off, Wade understands the difficult time most students have with the transition from the early calculus courses to the higher, more abstract classes. Several proofs in this text are a little more drawn out than in others, and this is to aid the in the understaning of the art of proof writing (and it really is an art).

Another think I liked is the variety of exercises. They range from relatively easy to rather difficult (I was unable to solve some...).

Now for the bad...

I really didn't like the layout of some of the material. Especially in chapter 1. He states a definition, and then it's results. Another definiton will be thrown in along with more results. Especially in chapter 1, it would be nice if all the definitions and axioms are put into one place, and the results to follow in a later section (Shilov does this, and I used his book a lot as well during the course). I find that referencing definitions that are organized in such a fashion much nicer.

If you're a budding mathematician, and would like to get a better understanding of analysis, I'd feel confident about adding this one to your collection.

Rating:
Summary: A Great Textbook for mathematics lover
Review: It has a lot of problems and concise anwsers and hints. It is a very good book to challeng your mind and push you to think more. I've read Apostol's, Rudin's and Courant John's Analysis textbook. Wade's is even easier and also clear.

Rating:
Summary: Wonderful book on Analysis
Review: One of the best and clearest book on analysis of all times. Modern introduction. I would like to have a solutions manual so it would be perfect for self study.

Rating:
Summary: Wonderful book on Analysis
Review: One of the best and clearest book on analysis of all times. Modern introduction. I would like to have a solutions manual so it would be perfect for self study.

Rating:
Summary: An irritating book.
Review: The strongest point of the book is the exercises. They force you to reread and understand the proofs and they build a foundation for material that is to come.

Chapter 1 (2nd edition) need a complete rewrite. How can you obfuscate something as simple as the Archimedean property in all its forms? Chapter 8 on Euclidean spaces needs to be better integrated with what the student should know from the first linear algebra course.

The author's proofs are not clear and I found myself rewriting many of them in my own words or turning to other references.

The core chapters 2-7 and 11-13 are fine - especially if you buy the approach of doing analysis first in R and then doing it a second time in R^n. This may be especially appropriate in an environment where most of the students are future high school teachers and will only take 1 advanced calculus course.

There are an unusual number of typos in the second edition. They are no longer accessible on the author's website. But hey, the 3rd edition is available, just throw out the 2nd and get the latest.