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Introduction to Analysis Paperback – 1 Dec 2007

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Product details

  • Paperback: 696 pages
  • Publisher: Pearson; 4 edition (1 Dec. 2007)
  • Language: English
  • ISBN-10: 0136153704
  • ISBN-13: 978-0136153702
  • Product Dimensions: 17.6 x 2.3 x 23.6 cm
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Bestsellers Rank: 966,172 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Product Description

About the Author

William Wade received his PhD in harmonic analysis from the University of California―Riverside. He has been a professor of the Department of Mathematics at the University of Tennessee for more than forty years. During that time, he has received multiple awards including two Fulbright Scholarships, the Chancellor's Award for Research and Creative Achievements, the Dean's Award for Extraordinary Service, and the National Alumni Association Outstanding Teaching Award.


Wade’s research interests include problems of uniqueness, growth and dyadic harmonic analysis, on which he has published numerous papers, two books and given multiple presentations on three continents. His current publication, An Introduction to Analysis,is now in its fourth edition.


In his spare time, Wade loves to travel and take photographs to document his trips. He is also musically inclined, and enjoys playing classical music, mainly baroque on the trumpet, recorder, and piano.

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Format: Paperback Verified Purchase
I admit, it's quite a while since i used this book. Also, I don't remember mistakes, typos, or non-sense. However, I thought it to be rather heavy on theorems and lacking clear examples a bit. I didn't find any better substitute at the time, so maybe that just the nature of the beast. I feel I lacked some more graphic, explanatory sections, describing the nature and implications of the theorems however.
I don't know how others feel about it, but this book wasn't really for me.
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By Shuo Dai on 17 Dec. 2014
Format: Paperback Verified Purchase
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Most Helpful Customer Reviews on (beta) 28 reviews
16 of 16 people found the following review helpful
Please take the other reviews with a grain of salt.. 16 Nov. 2012
By Eric - Published on
Format: Paperback
One thing that should be understood is that this book is used in many university advanced calculus courses. Advanced calculus courses, in (almost) all universities, have very high dropout rates, regardless of the textbook used. This is because advanced calculus is a tough course. Often students take a very soft (i.e., not very rigorous) introduction to mathematical proofs before advanced calculus. However, one must be able to read and write proofs very well before taking AC. Often students are not prepared, fail the course miserably, and blame the book, their teachers accent, or the professor of their "intro to proofs" class, and so on. Before taking advanced calculus I had worked through Spivak's Calculus and read other rigorous math books before taking this course, thus I could write proofs reasonably well and had few problems with this book, and neither will you if your proofing skills are in good shape.

That being said, the book is not bad, at all. It gets the job done. As far as rigor goes, very few things are assumed and almost everything that is used throughout the book is proved. Some of the few things that are not proved but are used throughout out the text are the continuity of functions such as sin(x) or e^x. The problems in the book are generally easy and not very interesting. I mean to say they do not illustrate the "beauty" of mathematical analysis. The amount proofs I found in the book to be "difficult to understand" I could count on one hand. When I encountered such a proof following along with pen and paper working out the details and filling in the gaps on my own would clear things up.

Rigor tends to be a common complaint among the other reviewers. Rigor is generally refers to the development of a theory, by essentially proving every little detail. Generally this means that nothing goes un proven; even if an assertion may be completely obvious, it is still formally proved. Not once in this book have I encounter a proof that lacked in rigor or used a certain technique or result that had not been discussed earlier in the book. Some have complained that the proofs are too long or too short, however compared to other analysis texts that I have seen, the proofs in this book seem very standard (although there are a few exceptions).

If your analysis class requires you to use this text, don't think you are doomed or will never be a successful mathematics student. If you can read and write a proof you will do fine, but I would not recommend this book for self study because there are much better, well established, classics in the field of Elementary Real Analysis. Such as "Calculus", by Michael Spivak, "Real Mathematical Analysis" by Charles C. Pugh, and finally, slightly dated but arguably the best, "Principles of Mathematical Analysis" by Walter Rudin.

Edit: If one is willing, going to the "All Comments" section and viewing the two comments above mine proves my point of students struggle with an Advanced Calculus course and take it out on their book. (The sad reality is that this book is much easier than the books I recommended above.)
12 of 12 people found the following review helpful
An irritating book. 4 Sept. 2004
By Charles R. Williams - Published on
Format: Hardcover
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.
8 of 10 people found the following review helpful
+ve multidimensional analysis -ve atrocious binding, 1-dimensional analysis could've been more detailed 18 Oct. 2005
By pineapple41 - Published on
Format: Hardcover
This book was used in my Analysis I class. I later had to prepare again for my masters certifying exam on Analysis and the primary reason I didn't use this book, even though I owned it, was the binding. By the end of my analysis course, it was practically in individual pages. The binding is atrocious - how can a $100+ book just be sort of glued together weakly that it falls apart after 1 semester of use.

The other reason I didn't use this book was that it goes through the 1-dimensional analysis pretty fast. My chosen analysis preparation book was also supposed to be my preparation for Royden's Real Analysis but I choose a book that covers the 1-dimensional analysis in twice the amount of pages and exercises taking a more detailed topology route. So, I'd advise a little bit of caution there if this is your path to graduate level Real Analysis.

However, with the 3rd edition out, our department threw out all the old 2nd edition books. I needed a multi-dimensional analysis text to prepare for my graduate PDE course and took 3 copies of the thrown ones out.

The reason I took 3 copies was that I knew the binding was going to fall apart and sure enough it did. It sort of breaks down into little booklets each that is glued to the spine. The pages on booklet peel off real easy and soon you have just pages instead of a book.

As for the multidimensional analysis it covers, it was very entertaining and fun to do. Chapter 13 becomes a little bit in the air as the exercises get a little tedious with calculate this integral with all sorts of surfaces enclosing it and not really much exercises requring a lot of threading of analysis ideas. I don't as of yet know if it has me prepared enough for Evan's PDE book but this was the only text conviniently showed up in the "free books" bin. But, I did have a nice linear and fun writing to it and I enjoyed it. Though not much as some of my other books.
Takes a challenging subject and makes it needlessly complicated 27 Nov. 2013
By Southern Man - Published on
Format: Paperback Verified Purchase
I loved my Intro to Analysis class, despite this book. The textbook took a challenging subject and made it needlessly complicated. There aren't enough examples and proofs, especially in the first two chapters (which is where they're needed most, because we're just starting).

After the first two chapters the proofs improve and the examples are marginally more plentiful, but the author struggles with definitions throughout the entire book. He usually does a decent job presenting theorems (with some big failures, like Theorem 4.3), but for some reason he can't define new terms in a clear way. He often fails to link the definition to his following example: The example doesn't explain how it uses the definition, so it doesn't help us understand the definition or how to use it. Sometimes the problem is a simple design/formatting issue: the "definition" is a single minimalist sentence enclosed in a text box, but information that is crucial to understanding the definition is on the previous page or in the next paragraph, not in the text box where it should be.

The most embarrassing gaffe is in Chapter 4.1 where the author "proves" Example 4.7 in a single line. The proof starts, "By the Power Rule (see Exercise 4.2.7), the answer is"... QED. But the Power Rule isn't introduced until the next chapter. Worse still, Exercise 4.2.7 is a homework problem that is left to the student, so you can't follow along and see how the author got his answer. I would love to use that method on a test: "Proof: By a theorem you don't know yet, the answer is 42. If you have questions, just refer to a more advanced problem you haven't seen yet. Once you solve that problem, use it to understand this basic concept."

Finally, the notation is confusing and makes it hard to study and organize ideas. Is Theorem 4.3 in Chapter 4.3? No! Theorem 4.3 is in Chapter 4.1, along with Example 4.7. How about the homework exercises - do they follow the same naming convention? Not at all! They start with Exercise 4.1.0.

OK, rant over. But professors, please consider using a different textbook. According to my professor, the 1st edition of Wade's book was very well written, but every edition since then has been a step backwards. I don't think they'll continue to use Wade in following years. Also, why so many editions? Has analysis changed that much, or is it just an excuse for Pearson to sell more overpriced textbooks?

I have Stephen Abbot's "Understanding Analysis" and I find it presents the material better - using intuitive arguments followed up by rigorous proofs. But it's not the text I used for the class, so I didn't spend as much time reading it - maybe it has other shortcomings. One thing Wade offers that Abbot doesn't is multi-dimensional theory. Still, it is worth investigating.

As far as other options, a number of reviews have presented alternative texts. Unfortunately, anyone who has had the opportunity to compare & contrast multiple Intro Analysis textbooks is probably not a beginner to real analysis, so their point of view might differ from a beginner's.

If I were a professor looking for a new Intro Analysis textbook, I'd bring a copy of each of the texts I liked to the prerequisite class for Intro to Analysis. If the students in that class can self-study chapter 1 of the textbook, you've found a winner.
2 of 3 people found the following review helpful
Messy Layout, Messy Ideas 25 Nov. 2009
By Kevin A. Brown - Published on
Format: Paperback
I used this for an undergraduate Analysis text for multivariate integration.

On the theory aspect, it was not entirely rigorous. Many fundamental ideas had to be found in asides randomly located in the text as "it can be shown" without justification. While this is true, when encountering the subject for the first time, the student might wish to at least have these facts readily available for reference, or explicitly included in the proofs. But worse, in the sections on vector calculus, the approach was explicitly non-rigorous, and any point requiring a rigorous explanation was relegated to "see Spivak's Calculus on Manifolds," a reference repeated several times throughout the later sections.

In terms of applications, there are plenty of computational examples and exercises. Unfortunately, the examples tend to be relatively easy compared with the exercises, and often rely on some trick which avoids the general principle to arrive at a quick solution. For instance, for parametrized curves, the examples involve things like connected line segments, while the exercises are much, much more complicated.

This attempt to balance rigor and applications unfortunately leaves the student confused and frustrated, not knowing the theoretical foundation or how to apply the results beyond toy problems. Just because a book has less rigor doesn't mean it's easier! It leaves the reader confused as to what rigorous proof even means, which is one of the key elements of introductory analysis classes.

That said, I haven't read the earlier chapters, and there may be something worthwhile inside them.
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