Introduction to Real Analysis Hardcover – Jan 18 2011
|New from||Used from|
Frequently Bought Together
Customers Who Bought This Item Also Bought
No Kindle device required. Download one of the Free Kindle apps to start reading Kindle books on your smartphone, tablet, and computer.
Getting the download link through email is temporarily not available. Please check back later.
To get the free app, enter your mobile phone number.
Top Customer Reviews
Most Helpful Customer Reviews on Amazon.com (beta)
Their treatment of limits, of continuity, of convergence, of differentiation and integration is exact and complete. They give readers a full grounding in epsilon/delta proof methodology for the major theorems of modern single variable calculus.
Because they deal in a single variable, they don't spend much time on basic topology. The book consists of eight chapters. A brief introduction to set theory is followed by a presentation of the real number system. Note that they don't construct the field of real numbers, they merely state the completeness theorem that fills in the gaps found in the field of rational numbers (e.g. the square root of two is a real number not found in the rationals).
The meat of the book begins with chapter three on sequences followed by chapters on limits & continuity, differentiation, Riemann integration, sequences of functions, and finally infinite series.
The many exercises will give readers much opportunity to hone their skills.
I have a few pet peeves. I find the tone a little patronizing. Walter Rudin's Principles of Mathematical Analysis is much more rigorous and explores the topic in greater depth than does Bartle & Sherbert's textbook, but he nowhere adopts their slightly consdescending tone.
Also, the presentation is a little dry. Many of the theorems they give are profound and exciting but one doesn't get this from the text. And they miss out on even hinting at fascinating results because it falls outside the scope of their program. For example, they spend a great deal of time on a rigorous elaboration the sine as cosine functions purely through their derivative properties, with no reference to their geometry interpretation. But because their text doesn't deal with complex numbers, they miss out on presenting a beautiful result that follows straightforwardly from this construction.
Overall, a solid and correct but not very inspiring introduction to the topic. Still, this is a great book from which to teach a course. Teachers can supply the inspiration themselves.
Vincent Poirier, Tokyo
The following chapters deal with sequences and series of numbers, limits, continuity, differentiation, integration, sequences and series of function, in this order.
I think the material is presented clearly and the results are proven rigorously throughout the entire book. There are a lot of worked-out examples and many exercises that will test the reader's understanding. Solutions and hints to many (notice, not only the odd ones) of the problems are given in the back of the book. There is also an appendix on logic for those who might need to review the basics, and one on metric spaces and Lebesgue integrals for those students who want to go a bit farther.
In my opinion, this book is not as good as Rudin's book, but it does the job better than many other introductory books on the same topic. For a horrible book see Jiri Lebl's text.
Real analysis is hard, independently of the book you use. It requires a lot of care and hard work. This book does the best it can at clearing the path for you.