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A Basic Course in Real Analysis, by Ajit Kumar, S. Kumaresan
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Based on the authors’ combined 35 years of experience in teaching, A Basic Course in Real Analysis introduces students to the aspects of real analysis in a friendly way. The authors offer insights into the way a typical mathematician works observing patterns, conducting experiments by means of looking at or creating examples, trying to understand the underlying principles, and coming up with guesses or conjectures and then proving them rigorously based on his or her explorations.
With more than 100 pictures, the book creates interest in real analysis by encouraging students to think geometrically. Each difficult proof is prefaced by a strategy and explanation of how the strategy is translated into rigorous and precise proofs. The authors then explain the mystery and role of inequalities in analysis to train students to arrive at estimates that will be useful for proofs. They highlight the role of the least upper bound property of real numbers, which underlies all crucial results in real analysis. In addition, the book demonstrates analysis as a qualitative as well as quantitative study of functions, exposing students to arguments that fall under hard analysis.
Although there are many books available on this subject, students often find it difficult to learn the essence of analysis on their own or after going through a course on real analysis. Written in a conversational tone, this book explains the hows and whys of real analysis and provides guidance that makes readers think at every stage.
- Sales Rank: #2805286 in Books
- Brand: Brand: Chapman and Hall/CRC
- Published on: 2014-01-10
- Original language: English
- Number of items: 1
- Dimensions: 9.40" h x .80" w x 6.10" l, 1.15 pounds
- Binding: Hardcover
- 322 pages
- Used Book in Good Condition
Review
"… there are some unique features that put this book aside. … a welcome addition to the library of teachers and student alike."
―Zentralblatt MATH 1308
"… this book describes the basic results of analysis in an extremely clear, straightforward, and well-motivated way. … if you’re looking for a text on the easy end of the spectrum for a course in real analysis, then this book is certainly worth a serious look …"
―MAA Reviews, October 2014
About the Author
Ajit Kumar has around 10 years of software development experience from embedded system to enterprise applications. He is an open source enthusiast who writes regularly for Linux For You (http: //www.linuxforu.com/) magazine about open source frameworks and tools.
Most helpful customer reviews
1 of 1 people found the following review helpful.
A good exposition with considerable errata
By B. Elam
I want to like this book and recommend it for a semester-long introduction to analysis. The exposition is good, but the other reviewer wasn't exaggerating about the errata. I'll start by elaborating on the positives, though. The authors forego depth of exposition in favor of taking time to help the reader develop intuition. Many proofs are prefaced by an equally long "Strategy" section that explains how one might go about constructing such a proof, which I think is great for an introductory text. The material presented should be at least as rewarding as less descriptive volumes are for students that take time to read the chapters beyond finding their assigned problem sets. Those that do will be well-equipped for a more advanced treatment of analysis.
It seems like the errors appear more often in definitions, statements of results, and exercises. If this were aimed at a more advanced audience, one might expect the reader to catch such errors more quickly. To give an example of what I consider a relatively minor error in the book, the authors characterize the density of the rational numbers in the real rile as, "Given a,b in R with a < b, there exists r in Q such that a < x < b." This is easy enough to catch and properly revise, but the errata becomes more subtle and confusing as the exposition grows more complex. Unfortunately, great exposition can't entirely make up for being consistently imprecise at the times when rigor is most necessary. I wouldn't recommend the text (in its present form) to anyone new to the subject, and I would recommend getting another text for reference to any student required to use this in their class.
Take heed of the text's "basic course" subtitle. Do not expect deep or varied coverage of integration. Its treatment is restricted to a chapter titled Riemannian Integration, with the following sections:
Darboux Integrability, Properties of the Integral, Fundamental Teorems of Calculus, Mean Value Theorems for Integrals, Integral Form of the Remainder, Riemann's Original Definition, Sum of an Infinite Series as an Integral, Logarithmic and Exponential Functions, and Improper Riemann Integrals.
That said, I very much enjoyed reading the book, and I would happily purchase a (better edited) second edition.
0 of 0 people found the following review helpful.
Errant, sloppy, not informative, not helpful
By D. I.
This book is riddled with mistakes. We had a boatload of cases where it would define a problem with one set of variables, then suddenly start using others, until after three changes it is extremely hard to keep up with. It is a first edition, but it makes me wonder if anybody even proofread it.
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