Elementary Analysis: The Theory of Calculus

by Kenneth A. Ross

Designed for students having no previous experience with rigorous proofs, this text can be used immediately after standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, as well as for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied, while many abstract ideas, such as metric spaces and ordered systems, are avoided completely. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics, and optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals.

Genres: Mathematics, Textbooks, Reference, Science, Nonfiction

272 pages, Hardcover

First published January 14, 1980

Reviews

[Umran Haji · Rating 5 out of 5]

This is a tremendous introduction to real analysis. Elitists seem to disparage this book in favor of the ubiquitous Rudin, yet I cannot fathom why. The biggest shortcoming of Ross relative to Rudin is that Rudin includes some more advanced content that makes it able to be used for a two-semester course. Ross ends at the Riemann-Stieltjes integral while Rudin goes beyond to other topics like vector analysis and measure theory. But there is nothing wrong with using Ross for a semester and then moving on to a different book when you are finished with it.

Almost everything in the first half of Rudin is in this book as well, including metric spaces, but Ross is careful and friendly in his expositions and tends to provide good motivation for the concepts. He meticulously references previous theorems throughout the book and you can really see how the concepts build on each other. Also, the difficulty of the book is just right. Personally, I already had sufficient "mathematical maturity" coming into my first course on real analysis, but I did not find this book "too easy", as some will claim. The first few exercises in each section are pretty plug-and-chug but the later problems require serious thought and usually lead to enlightening results. If you're using this for a class and your instructor chooses a good mix of problems, you will enjoy this book a lot, and you will find it a good reference when you later need to look up a theorem from basic analysis. Now that I am studying more advanced material, Ross has earned a permanent place on my bookshelf.

[Jonathan Keogh · Rating 5 out of 5]

This is one of the best introductions to rigorous mathematics! Looking back now, to when I was 18 and had no idea what the proper definition of a limit was, it was this book that gently eased me into proper college mathematics through a self-contained, purely proof-based, introduction to analysis. After this, I was able to pick up other more advanced books on my own, and continue my mathematical journey without anymore hand-holding.

I find it odd that one would disparage this book because it is too 'easy'. If it's too easy then it isn't for you! However, for one that is new to analysis and isn't a total genius, this text lucidly and clearly leads one through the subject, with plenty of motivation and careful reasoning.

Any layman that is interested in mathematics, and is willing to study hard on their own, should pick this up and do as much exercises as possible.

[Jake]

I’ve earned this

[Adam · Rating 3 out of 5]

This is not a bad introduction to the subject. My favorite feature of the book is that it is so efficient to the point and big picture of Analysis. However, I have found a few frustrating typos that have confused students, and proofs not always given in the clearest possible way. Perhaps worse than that is that the book omits much for the sake of efficiency. That's makes it good as book in-between the baking-recipe style James Stewart Calculus type texts and the much more mature Rudin styled Analysis texts, but probably shouldn't be what Math majors use to qualify as "having had a class in advanced Calculus / Analysis". I still don't have a book that I thoroughly like for that purpose though.

[Todd (theRealChainman) Chaney · Rating 2 out of 5]

I don't remember quite well cause I read the book 6 months ago for my summer Real Analysis course. I know that I had to supplement it by reading other textbooks in order to get a better understanding of the mathematical concepts. It felt like the book just provided you the information but lacked explaining the concept.

[Ben Zhang · Rating 4 out of 5]

a pretty dense read but not as dense as rudin of course - read this book while following along in a real analysis course and decided to finish read the rest of the book. if you've taken a higher-level calculus course then most of this may seem obvious already but sometimes it's good to whack your head trying to prove the obvious.. some simple theorems still had nontrivial proofs : )

[Aaron Stewart · Rating 5 out of 5]

This is a great textbook, and with the right professor, this leads nicely into Rudin, IMHO. I would use this in concord with Abbot as well, I felt properly prepared to tackle harder portions of analysis

[Saman Givian · Rating 5 out of 5]

An extremely well-written textbook in which everything is very crystal clear to understand.

[almond croissant · Rating 3 out of 5]

better than my prof

[Jingyuan · Rating 3 out of 5]

Easy to understand for a first course in analysis, but topology section is a mess, use Pugh or Rudin for topology sections instead.

[Nicholas Cotton · Rating 4 out of 5]

It’s definitely a math book, but it’s easier to read than other analysis books I’ve seen

I really appreciated the back of the book giving hints to exercises rather than full solutions.

[Xiaojie Johan · Rating 5 out of 5]

Great, easy and simple reading on elementary analysis!

[m]

高校の頃, 一時期ゼミで使ってたっけな.