| Title: | Understanding Analysis |
| Authors: | Abbott, Stephen |
| Keywords: | Mathematics Understanding Analysis |
| Issue Date: | 2016 |
| Publisher: | Springer |
| Abstract: | My primary goal in writing Understanding Analysis was to create an elemen- tary one-semester book that exposes students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and im- prove mathematical intuition rather than to verify it. There is a tendency, however, to center an introductory course too closely around the familiar the- orems of the standard calculus sequence. Producing a rigorous argument that polynomials are continuous is good evidence for a well-chosen definition of con- tinuity, but it is not the reason the subject was created and certainly not the reason it should be required study. By shifting the focus to topics where an untrained intuition is severely disadvantaged (e.g., rearrangements of infinite series, nowhere-differentiable continuous functions, Cantor sets), my intent is to bring an intellectual liveliness to this course by offering the beginning student access to some truly significant achievements of the subject. |
| Description: | This book is an introductory text. The only prerequisite is a robust understand- ing of the results from single-variable calculus. The theorems of linear algebra are not needed, but the exposure to abstract arguments and proof writing that usually comes with this course would be a valuable asset. Complex numbers are never used. The proofs in Understanding Analysis are written with the beginning student firmly in mind. Brevity and other stylistic concerns are postponed in favor of including a significant level of detail. Most proofs come with a generous amount of discussion about the context of the argument. What should the proof entail? Which definitions are relevant? What is the overall strategy? Whenever there is a choice, efficiency is traded for an opportunity to reinforce some previously learned technique. Especially familiar or predictable arguments are often deferred to the exercises. The search for recurring ideas exists at the proof-writing level and also on the larger expository level. I have tried to give the course a narrative tone by picking up on the unifying themes of approximation and the transition from the finite to the infinite. Often when we ask a question in analysis the answer is “sometimes.” Can the order of a double summation be exchanged? Is term-by- term differentiation of an infinite series allowed? By focusing on this recurring pattern, each successive topic builds on the intuition of the previous one. The questions seem more natural, and a coherent story emerges from what might otherwise appear as a long list of theorems and proofs. This book always emphasizes core ideas over generality, and it makes no effort to be a complete, deductive catalog of results. It is designed to capture the intellectual imagination. Those who become interested are then exceptionally well prepared for a second course starting from complex-valued functions on more general spaces, while those content with a single semester come away with a strong sense of the essence and purpose of real analysis. |
| URI: | http://localhost:8080/xmlui/handle/123456789/217 |
| ISBN: | 978-1-4939-2712-8 |
| Appears in Collections: | ARTS & SCIENCE |
| File | Description | Size | Format | |
|---|---|---|---|---|
| 2015_Book_UnderstandingAnalysis.pdf | 3.49 MB | Adobe PDF | View/Open |
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