Real Analysis: A Constructive Approach 1st Edition by Mark Bridger (PDF)

Description

A unique approach to analysis that lets you apply mathematics across a range of subjectsThis innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. In particular, existence is never established by showing that the assumption of non-existence leads to a contradiction. The ultimate consequence of this method is that it makes sense—not just to math majors but also to students from all branches of the sciences.The text begins with a construction of the real numbers beginning with the rationals, using interval arithmetic. This introduces readers to the reasoning and proof-writing skills necessary for doing and communicating mathematics, and it sets the foundation for the rest of the text, which includes:Early use of the Completeness Theorem to prove a helpful Inverse Function TheoremSequences, limits and series, and the careful derivation of formulas and estimates for important functionsEmphasis on uniform continuity and its consequences, such as boundedness and the extension of uniformly continuous functions from dense subsetsConstruction of the Riemann integral for functions uniformly continuous on an interval, and its extension to improper integralsDifferentiation, emphasizing the derivative as a function rather than a pointwise limitProperties of sequences and series of continuous and differentiable functionsFourier series and an introduction to more advanced ideas in functional analysisExamples throughout the text demonstrate the application of new concepts. Readers can test their own skills with problems and projects ranging in difficulty from basic to challenging.This book is designed mainly for an undergraduate course, and the author understands that many readers will not go on to more advanced pure mathematics. He therefore emphasizes an approach to mathematical analysis that can be applied across a range of subjects in engineering and the sciences.

User’s Reviews

Editorial Reviews: Review “The first chapters are presented at a very nice leisurely pace, which makes reading and learning enjoyable.” (Zentralblatt MATH, 2007) “Very suitable for self-study by undergraduates at all levels…” (CHOICE, August 2007)”…deserves to be read. Even if you do not subscribe to the constructive viewpoint, you’ll learn something and find plenty of material to exploit in your classical analysis courses.” (MAA Reviews, December 23, 2006) From the Back Cover A unique approach to analysis that lets you apply mathematics across a range of subjectsThis innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. In particular, existence is never established by showing that the assumption of non-existence leads to a contradiction. The ultimate consequence of this method is that it makes sense—not just to math majors but also to students from all branches of the sciences.The text begins with a construction of the real numbers beginning with the rationals, using interval arithmetic. This introduces readers to the reasoning and proof-writing skills necessary for doing and communicating mathematics, and it sets the foundation for the rest of the text, which includes:Early use of the Completeness Theorem to prove a helpful Inverse Function TheoremSequences, limits and series, and the careful derivation of formulas and estimates for important functionsEmphasis on uniform continuity and its consequences, such as boundedness and the extension of uniformly continuous functions from dense subsetsConstruction of the Riemann integral for functions uniformly continuous on an interval, and its extension to improper integralsDifferentiation, emphasizing the derivative as a function rather than a pointwise limitProperties of sequences and series of continuous and differentiable functionsFourier series and an introduction to more advanced ideas in functional analysisExamples throughout the text demonstrate the application of new concepts. Readers can test their own skills with problems and projects ranging in difficulty from basic to challenging.This book is designed mainly for an undergraduate course, and the author understands that many readers will not go on to more advanced pure mathematics. He therefore emphasizes an approach to mathematical analysis that can be applied across a range of subjects in engineering and the sciences. About the Author MARK BRIDGER, PHD, is Associate Professor of Mathematics at Northeastern University in Boston, Massachusetts. The author of numerous journal articles, Dr. Bridger’s research focuses on constructive analysis, the philosophy of science, and the use of technology in mathematics education. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I’m a mathematics undergraduate at a fairly mediocre (but ‘up-and-coming’) state university in the US. The analysis textbook we currently use (like most of those I’ve seen) has literally no discussion at all of the ‘foundational issues’ around the so-called Real Numbers. It’s like it’s stuck in the 19th century, before anybody ever started thinking seriously about computation; the Completeness Axiom they give us is a tough pill to swallow for someone acquainted with 20th-century undecidability results. When I asked my profs if we’ll ever get around to constructing the real line, Dedekind-style, they told me it’s “too advanced”. Bridger shows how wrong they are about that! This is the most accessible, modern, and relevant introductory analysis textbook I’ve found yet. It’s a pleasure to read, and Bridger’s constructive version of the reals, via his Completeness Theorem, is something I can actually believe in. It’s not crammed full of theorems to memorize, but instead gently and consistently develops a student’s proof-writing ability and intuition. If I had the money, I’d buy a copy for everyone on the faculty at my school.

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