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Real Analysis, by Frank Morgan
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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).
- Sales Rank: #1473434 in Books
- Brand: Brand: American Mathematical Society
- Published on: 2005-08-01
- Released on: 2005-07-09
- Original language: English
- Number of items: 1
- Dimensions: 10.00" h x 7.25" w x .50" l, 1.15 pounds
- Binding: Hardcover
- 151 pages
- Used Book in Good Condition
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
Most helpful customer reviews
10 of 10 people found the following review helpful.
Perfect start beyond linear algebra.
By 05naa
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.
4 of 4 people found the following review helpful.
For beginners and a quick review for not so beginners
By Javier AP
This is no conventional Analysis book. It is written to drive the reader's focus to a single topic each time. The chapters are 1-2 pages long and it is useful for what is is thought for. Continuity? You read 1 page, straight to the point. Facts about number sets and infinity? 1 page. Obviously it leaves out many consequences, but the uniqueness of this book is that you will always find the basic facts you need right in the first page you happen to open. It is written so concisely that you will understand everything in the first read. In fact, I like to refer to it as "my first real analysis book". OK, it is true that maybe it is too much to ask $40 for it, but is definitely nice to have it around.
6 of 7 people found the following review helpful.
perfect for what it's written for
By apostrophe
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.
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