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Basic Real Analysis, by Anthony W. Knapp
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Systematically develop the concepts and tools that are vital to every mathematician, whether pure or applied, aspiring or established
A comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics
Included throughout are many examples and hundreds of problems, and a separate 55-page section gives hints or complete solutions for most.
- Sales Rank: #1487863 in Books
- Brand: Brand: Birkhäuser
- Published on: 2005-07-29
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.50" w x 6.14" l, 2.33 pounds
- Binding: Hardcover
- 650 pages
- Used Book in Good Condition
Review
From the reviews:
“The volume contains more than 300 problems and a separate section gives hints or complete solutions to them. The book seems to be completely unified, carefully reasoned, rich in concepts, methods and results, and indubitably useful as for students in Real Analysis so also for teachers in this field.”(Zentralblatt MATH)
"This book tries to develop concepts and tools in real analysis that are vital to every mathematician. … The book contains more than 300 problems with hints and complete solutions for many of them." (A. Kriegl, Monatshefte für Mathematik, Vol. 151 (3), 2007)
From the Back Cover
Basic Real Analysis and Advanced Real Analysis (available separately or together as a Set) systematically develop those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. These works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics.
Key topics and features of Basic Real Analysis:
* Early chapters treat the fundamentals of real variables, sequences and series of functions, the theory of Fourier series for the Riemann integral, metric spaces, and the theoretical underpinnings of multivariable calculus and differential equations
* Subsequent chapters develop the Lebesgue theory in Euclidean and abstract spaces, Fourier series and the Fourier transform for the Lebesgue integral, point-set topology, measure theory in locally compact Hausdorff spaces, and the basics of Hilbert and Banach spaces
* The subjects of Fourier series and harmonic functions are used as recurring motivation for a number of theoretical developments
* The development proceeds from the particular to the general, often introducing examples well before a theory that incorporates them
* The text includes many examples and hundreds of problems, and a separate 55-page section gives hints or complete solutions for most of the problems
Basic Real Analysis requires of the reader only familiarity with some linear algebra and real variable theory, the very beginning of group theory, and an acquaintance with proofs. It is suitable as a text in an advanced undergraduate course in real variable theory and in most basic graduate courses in Lebesgue integration and related topics. Because it focuses on what every young mathematician needs to know about real analysis, the book is ideal both as a course text and for self-study, especially for graduate students preparing for qualifying examinations. Its scope and approach will appeal to instructors and professors in nearly all areas of pure mathematics, as well as applied mathematicians working in analytic areas such as statistics, mathematical physics, and differential equations. Indeed, the clarity and breadth of Basic Real Analysis make it a welcome addition to the personal library of every mathematician.
Most helpful customer reviews
21 of 24 people found the following review helpful.
Great book for real analysis
By Fadi
This is a fabulous book for learning real analysis. I've never taken a course in the subject (High school doesn't have one) but I have used the book for an independent study. After a fair study of point-set topology, this book can take you in and out of real analysis with ease. It's comprehensive and provides interesting insight into fields you thought you knew. It first tackles the problem of axiomatizing basic calculus, then it gives some background on metric spaces before hitting calculus of several variables and ordinary differential equations. The chapter on metric spaces was far more comprehensive than any I've read, and I've never seen a book that develops the theory of differential equations from an abstract perspective (none to this degree at least).
After this, it hits the meat of real analysis: Measure theory (and Lebesgue integration). Again, the author does an excellent job explaining and elaborating on this powerful theory. Euclidean spaces, topological spaces, L^p spaces, and Hilbert/Banach spaces take up the rest of this book. It even includes a very interesting chapter on Fourier transforms for Euclidean spaces. Overall, it's definitely worth the buy.
0 of 2 people found the following review helpful.
Clunky, needs editing....
By Nicholas Stevenson
This text is very thorough, and has many exercises with solutions and hints for solutions.
BUT.... it needs editing before I would use it as a course textbook for students learning this subject, as the presentation seems more tangled and turgid than necessary; I suspect that the author took the text material straight from his old notes. For example, on p. 578, Knapp gives the long, difficult proof of the Cantor-Schroeder-Bernstein theorem, instead of Koenig's short, beautiful, memorable proof from 1906). Or look at Corollary 1.5 on p.4. The last sentence of this proof is completely redundant as (n+1)/N from the previous sentence is a rational number between x and y.
Overall, a good text but Rudin is still better.
12 of 33 people found the following review helpful.
Kindle edition is unreadable
By Dave
The kindle edition has many unreadable equations in it.
I don't know the root cause (translation to kindle format, etc.)
The bottom line is some of the equations are worthless.
Don't buy this book in this format.
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