Real Analysis [Hardcover]

Frank Morgan (Author)

Book Description

ISBN-10: 0821836706 | ISBN-13: 978-0821836705 | Publication Date: July 9, 2005

This book is written by award-winning author, Frank Morgan. It offers a simple and sophisticated point of view, reflecting Morgan's insightful teaching, lecturing, and writing style. Intended for undergraduates studying real analysis, this book builds the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in $\mathbb{R}^n$. It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets. It gives the three characterizations of compactness: as "closed and bounded," via sequences, and via open covers. Topics include Fourier series, the Gamma function, metric spaces, and Ascoli's Theorem. This concise text not only provides efficient proofs, but also shows students how to derive them. The excellent exercises are accompanied by select solutions. Ideally suited as an undergraduate textbook, this complete book on real analysis will fit comfortably into one semester. Frank Morgan received the first Haimo Award for distinguished college teaching from the Mathematical Association of America. He has also garnered top teaching awards from Rice University (Houston, TX) and MIT (Cambridge, MA).

Editorial Reviews

Review

"Reading your book is a refreshingly delightful change from the usual emphasis on series, rather than topology, as a foundation of analysis." ---- Robert Jones, University of Dusseldorf

Product Details

• Hardcover: 151 pages
• Publisher: American Mathematical Society (July 9, 2005)
• Language: English
• ISBN-10: 0821836706
• ISBN-13: 978-0821836705
• Product Dimensions: 10.2 x 7.2 x 0.6 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.5 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #660,258 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

6 of 6 people found the following review helpful:

5.0 out of 5 stars Perfect start beyond linear algebra., July 27, 2006

By 

05naa "Nas" (Boston, MA) - See all my reviews

This review is from: Real Analysis (Hardcover)

This book is designed to be a one semester book in undergrad analysis.

It covers almost all the material in either Rudin's or Krantz's, which are usually covered in two semesters. however, Prof. Morgan does sacrafice some of the depth that other texts go into (example: theorem a sequence is convegent <=> cauchyis only proved the forward way).

Why I like it:
-This approach is perfect. You get a solid underpinning in the basics and get a taste of many diffrent subjects, (fourier, stirling's formula, Volumes of N-balls etc...). It is perfect for a first class in upper math (beyond linear algebra).

-the problems sets are actual problem sets that should be done on a per lecture basis. other books are filled with extensive problems that usually overwhelm a student with only a linear algebra background.

-all the proofs relating to compactness are written with the three equivilent def. of compactness (closed & bounded in R^N, open covers, sequential compactness). This allows a student to fully appreciate the modern defenition of open covers for compactness.
the same goes for proofs involving continuity (sequential, epsilon-selta, inverse images of opens sets are open).
Both of these techniques makes the transition to general metric spaces, where the open covers and inverse images are primaraly used, smooth.

3 of 3 people found the following review helpful:

5.0 out of 5 stars perfect for what it's written for, January 16, 2008

By 

apostrophe - See all my reviews

This review is from: Real Analysis (Hardcover)

This book is simply beautiful. It is written as a textbook for just one semester of undergraduate study, and it does the job. It is concise, jet it contains all the main theorems with the proofs rigorous enough. It begins with elementary concepts of numbers and logic, and even topology (so it treats continuity via open sets as well!), compactness is treated in three different ways, Fourier series is explained, and it even touches metric spaces... The book doesn't go very deep, but it talks well about basic issues in analysis and it's rigorous enough for an undergraduate text. The book can even serve well as a manual for those who need to refresh their knowledge about main theorems of analysis while considering other fields of mathematics invoking some analysis issues, because it's concise, rigorous enough and well organized. Very useful.

9 of 12 people found the following review helpful:

1.0 out of 5 stars Real Analysis for preschoolers, April 16, 2006

By 

Kate (Iowa City, IA) - See all my reviews

This review is from: Real Analysis (Hardcover)

This is a truly awful textbook. It is written in a patronizing and demeaning tone, suggesting the author does not consider his audience actually capable of learning real analysis. The chapters are brief, often only a single page, and seem to barely scratch the surface of the topic. Many proofs are left out altogether, with occasional suggestions that the reader "find the proof on the internet." Important concepts are also excluded or mentioned only in problems, such as the definition of a Cauchy sequence. The section on Fourier Series includes misprints in two essential definitions. The book is almost entirely void of examples, and often seems to do little more than merely list theorems. Although the price is much less than many other introductory texts, it would be worth your money to look elsewhere.