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Introduction to Analysis Reviews

Customer Rating:
Summary: Analysis in the trenches
Comment: Mattuck's 400-page book (almost entirely single variable except for differentiation under the integral and double integrals up to one iterate being improper) does not deliver more topics and deeper results as books half its size. The second mean value theorem used for deeper results in Fourier series convergence and integral inversion is not covered as in books by Apostol and Stromberg. The Arzela-Ascoli and Wierstrass Thms are not covered as in the 150-page Goffman book. Limsup and liminf takes up only half a page. Instead the book is rambling and uses uncommon notations and definitions. Mattuck defines and uses widely the wavy equal sign in place of most statements on limits. The Riemann integral is defined differently then other books. The linearity of the integral takes two chapters to develop. Topology is not used in the significant way as in Stromberg, Rudin, and Rosenlicht to prove the intermediate value thm and the thm on the existence of max and min points in a compact domain using the preservation of connectedness and compactness by continuous maps. The book gets mired throughout in approximations and upper and lower bounds. Yet, the axioms of a complete ordered field (COF)is never stated even though Mattuck admits some good mathematicians trip over inequalities. Therefore, that all the fuss made in analysis is to show that the geometric and intuitive notions of calculus can be based on sets and the COF never comes through.

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Customer Rating:
Summary: An Excellent Bridge to Theoretical Math for Scientists
Comment: I had trained as a physicist in college, and found this book useful when I began my PhD work in mathematics, where the way of thinking was just different enough to trip me up. My real analysis and Lebesgue integration class used the formidable classic "Introductory Real Analysis" by Kolmogorov and Fomin, and this book was a useful adjunct as I worked through the material for the class.

The authors take a more explicitly numerical or equation-based approach to analysis than Kolmogorov and Fomin, who are more abstract and set-based. While this book does not go into topic Lebesgue integrals with anywhere near the depth of K&F, the appeal to numerical thinking is useful for helping someone in natural science get a handle on where the abstract math is going. When I read the authors' introduction, I was gratified to know that this book's approach stemmed from the travails that physics majors at MIT faced when they took real analysis!

I found this book handy for the basics on the limits of sets and the Picard condition for ordinary differential equations. Its coverage of more advanced topics like Lebesgue integration is very light, but as an undergraduate text or as an adjunct for graduate students new to the field, it can be highly useful.

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Customer Rating:
Summary: Newest of the Best!
Comment: I have always been hoping that there are more modern textbooks on the undergraduate math that are written by great mathematicians. This one is one of the first to appear on the market. Thanks for Professor Mattuck! I love you, and love your book more!!! Hehe....Oh, by the way, one day if you see a movie called A Wonderful Mind, then you'll find a guy featured in it whose name is Abel Maple. The future greatest mathematician of the 22 century. Just kidding!

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Customer Rating:
Summary: Great introduction to analysis
Comment: The book is slow to begin but it does a great job in explaining all the concepts. The author explains the proofs and theorems and it introduces some intermediate ideas to understand the theorems and definitions. The book contains a lot of exercise of different nature and difficulty. It covers a great range of subjects but not enough on the Rn. The book is basic in it contain, it is not difficult to read and follow. It can serve as an introduction to analysis. I would recommend it if you want an introduction to analysis.

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Customer Rating:
Summary: A great introduction to analysis
Comment: This is an unusual and beautifully written introduction to real analysis. The presentation is carefully crafted and extremely lucid, with wonderfully creative examples and proofs, and a generous sprinkle of subtle humor. The layout of the pages is exceptionally attractive. The author has clearly put a great deal of thought and effort into producing an analysis text of the highest quality.

Most of the book concentrates on real-valued functions of a single (real) variable. There is a gradual and careful development of the ideas, with helpful explanations of elementary matters that are often skipped in other books. For instance, prior to the chapter on limits of sequences, the book has a chapter on estimation and approximation, discussing algebraic laws governing inequalities, giving examples of how to use these laws, and developing techniques for bounding sequences and for approximating numbers. Proofs involving "epsilons" and "arbitrarily large n" make their first appearance here.

The overall presentation of the book is carefully thought out. Each chapter is broken up into small sections, and each section emphasizes one principle idea or theorem. The proofs of the main theorems are lovely, and give both intuitive explanations and rigorous details. Genuinely interesting examples and problems illuminate the key ideas. Each chapter contains a mix of problems: "questions" that help students test their grasp of the main points of each section, "exercises" that are intermediate in scope, and more difficult "problems". (A solutions manual is available for instructors from the publisher.)

The careful explanations, even of "elementary" matters, and two appendices on sets, numbers, logic, and methods of argumentation, make the book suitable for a first analysis course in which students have had no prior exposure to proofs. There is ample material for a one-semester, or in some cases a one-year, course.

In summary, I believe that this is the best introductory real analysis book on the market. Students and instructors alike will find it a joy to read.

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