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Understanding analysis
Stephen Abbott TNIDERSTANDING ANALYSIS Pp ringer Preface 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 subjcct was crcated and certainly not thc 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, Fourier series ), my intent is to restore an intellectual liveliness to this course by offering the beginning student access to some truly significant achievements of the subject The main Obiectives In recent years, the standard undergraduate curriculum in mathematics has been subjected to steady pressure from several different sources. As computers and technology become more ubiquitous, so do the areas where mathematical t hinking can be a valuable asset. Rather than preparing themselves for graduate study in pure mathematics, the present majority of mathematics majors look forward to careers in banking, medicine, law, and numerous other fields where analytical skills are desirable. Another strong influence on college mathemat ics is the ongoing calculus reform effort, now well over ten years old. At the corc of this movemcnt is the justifiable goal of presenting calculus in a morc tuitive way, emphasizing geometric arguments over symbolic ones. Despite these various trends or perhaps beca. Ise of them--nearly every undergraduate mathematics program continues to require at least one semester of real analysis The result is that instructors today are faced with the task of teaching a diffi- cult, abstract course to a more diverse audience less familiar with the nature of axiomatic arguments The crux of the matter is that any prevailing sentiment in favor of marketing mathematics to larger groups must at some point be reconciled with the fact Preface that theoretical analysis is extremely challenging and even intimidating for some One unfortunate resolution of this dilemma has been to make the course easier y making it less interesting. The omitted material is inevitably what gives analysis its true Favor. a better solution is to find a way to make the more advanced topics accessible and worth the effort I see three essential goals that a semester of real analysis should try to meet 1. Students, especially thiose emerging fron a reforIn approach to calculus need to be convinced of t, he need for a. more rigorous study of functions The necessity of precise definitions and an axiomatic approach must be carefully motivated aving seen mainly graphical, numerical, or intuitive arguments, students need to learn what constitutes a rigorous mathematical proof and how to write one 3. There needs to be significant reward for the difficult work of firming up the logical structure of limits. Specifically, real analysis should not be just, an elaborate reworking of standard introductory calculus. Students should be exposed to the tantalizing complexities of the real line, to the subtleties of different flavors of convergence and to the intellectual delights hidder in the paradoxes of the infinite The philosophy of Under standing ATalysis is to focus attentiOn Onl questions that give analysis its inherent fascination. Does the Cantor set contain any irrational numbcrs? Can the sct of points whcrc a function is discontinuous be arbitrary? Are derivatives continuous? Are derivatives integrable? Is an infinitely differentia ble function necessarily t he limit of its Taylor series? In giving these topics center stage, the hard work of a rigorous study is justified the fact that the cble without it The structure of the book This book is an introductory text. Although some fairly sophisticated topics are brought in early to advertise and motivate the upcoming material, the main body of each chapter consists of a lean and focused treatment of the core top ics that make up the center of most courses in analysis. Fundamental results about completeness, compactness, sequential and functional limits, continuity uniform convergence, differentiation, and integration are all incorporated. What is specific here is where the enphasis is placed. In the chapter On integratioN for instance, the exposition revolves around deciphering the relationship be twcon continuity and the Riemann integral. Enough propcrtics of thc integral are obtained to justify a proof of the Fundamental Theorem of Calculus, but the theme of the chapter is the pursuit of a characterization of integrable func tions in terms of continuity. Whether or not Lebesgue's measure-zero criterion is treated, framing the material in this way is still valuable because it is the questions that are important. Mathematics is not a static discipline. Students Preface should be aware of the historical reasons for the creation of the mathematics they are learning and by extension realize that there is no last word on the of integrati explicitly by including some relatively recent developments on the generalized Riemann integral in the additional topics of the last chapter The structure of the chapters has the following distinctive features Discussion Sections: Each chapter begins with the discussion of some mo- tivating examples and open questions. The tone in these discussions is inten tiOnally informal, anld full use is Illade of familiar functiOns alld results froll caLculus. The idea is to freely explore the terrain, providing context. for the upcoming definitions and theorems. A recurring theme is the resolution of the paradoxes that arise when operations that work well in finite settings are naively extended to infinite settings(e. g, differentiating an infinite series term-by-term reversing the order of a double summation). After these exploratory introduc- tions. the tone of the writing changes, and the treatment becomes rigorously tight but still not overly formal. With the questions in place, the need for the ensuing development of the material is well-motivated and the payoff is in sight Project Sections: The penultimate section of each chapter(the final section is a short cpiloguc) is written with the cxcrciscs incorporated into thc cxposition Proofs are outlined but not completed, and additional exercises are included to elucidate the material being discussed. The point of this is to provide some Flexibility. The sections are written as self-guided tutorials, but they can also be the subject of lectures. I have used them in place of a final examination and they work especially well as collaborative assignments that can culminate in a class presentation. The body of each chapter contains the necessary tools, so there is some satisfaction in letting the students use their newly acquired skills to ferret out for theirselves ans wers to questions that have beel driving the exposition Building a Course Teaching a satisfying class inevitably involves a race against time. Although this book is designed for a 12-14 week semester, there are still a few choices to make as to what to cover The introductions can be discussed, assigned as reading, omitted, or sub stituted wit h something preferable. There are no theorems proved here that show up later in the text. I do develop some important examples in these introductions(the Cantor set, Dirichlet's nowhere-continuous func tion)that probably need to find their way into discussions at some point Chapter 3, Basic Topology of R, is much longer than it needs to be. All that is required by the ensuing chapters are fundamental results about open and closed sets and a thorough understanding of sequential com pactness. The characterization of compactness using open covers as well Preface as the section on perfect and connected sets are included for their own in trinsic interest. They are not, however, crucial to any future proofs. The one exception to this is a presentation of the Intermediate Value Theorem (IVT) as a special case of the preservation of connected sets by continu ous functioNs. To keep connectedness truly optional, I have included two direct proofs of IVT, one using least upper bounds and the other usin ncstcd intervals. A similar comment can bc madc about pcrfcct scts. A though proofs of the Baire Category Theorem are nicely motivated by the argument that perfect, sets are uncountable, it is certainly possible to do ithout the othe All the project sections (1.5, 2.8, 3.5, 4.6, 5.4, 6.6,7.6,8.1-8.4)are optional in the sense that no results in later chapters depend on material in these sections. The four topics covered in Chapter 8 are also written in this project-style format, where the exercises make up a significant part of the development. The only one of these sections that might require a lecture is the unit on Fourier series, which is a bit longer than the others The audience The only prerequisite for this course is a robust understanding 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 coursc would be a valuable assct. Complcx numbcrs arc ncver uscd in this book The proofs in Understanding Analysis are written with the introductory student firmly in mind. Brevity and othcr stylistic concerns arc postponcd in favor of including a significant level of detail. Most proofs come with a fair amount of discussion about the context of the argument. What should the proof entail? Which definitions are relevant? What is the overall strategy Is one particular proof similar to something already done? Whenever there is a choice, efficiency is traded for an opportunity to reinforce some previousl learned technique. Especially familiar or predictable arguments are usually sketched as exercises so that students can participate directly in the development of the core material 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 b picking up on the unifying theine of approxilllation anld the tranisition fronn the finite to the infinite. To paraphrase a passage from the end of the book, rea numbers arc approximated by rational oncs: valucs of continuous functions arc approximated by values nearby, curves are approximated by straight lines; areas are approximated hy sums of rectangles; continuous functions are a..ted by polynomials. In each case, the approximating objects are tangible and well understood, and the issue is when and how well these qualities survive the limiting process. By focusing on this recurring pattern, each successive topic Preface builds on the intuition of the previous one. The questions seem more natural, and a method to the madness 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 Allure general spaces, while those content with a single seMester come away with a strong sense of the essence and purpose of real analysis. Turning once more to the concluding passages of Chapter 8, "By viewing the different infinities of mathematics through pathways crafted out of finite objects, Weierstrass and the other founders of analysis created a paradigm for how to extend the scope of mathematical exploration deep into territory previously unattainable This exploration has constituted the major thrill of my intellectual life. I am extremely pleased to offer this guide to what I feel are some of the most impressive highlights of the journey. Have a wonderful trip Acknowledgments The genesis of this book came from an extended series of conversations with Benjamin Lotto of Vassar College. The structure of the early chapters and the book's overall thesis are e part the result of several years of sharing classroom notes, ideas, and experiences with Ben. I am pleased with how the manuscript has turned out, and I have no doubt that it is an immeasurably better book because of Ben's early contributions A large part of the writing was done while I was enjoying a visiting position at the University of virginia. Special thanks go to Nat Martin and Larry Thomas for being so generous with their tilne anld wisdoN, alld especially to Lorel Pitt, the scope of whose advice extends well beyond the covers of this book. I would also like to thank julie riddleberger for her help with many of the figures Marian robbins of Bellarmine College, Steve Kennedy of Carleton College, Paul Hum ke of saint Olaf College, and tom Kriete of the university of virginia. each taught from a preliminary draft of this text. I appreciate the many suggested improvements that this group provided, and I want to especially acknowledge Paul Humke for his contributions to the chapter on integration My department and the administration of Middlebury College have also been very supportive of this endeavor. David Guertin came to my technologi cal rescue on IluImlerous occasiOns, Priscilla Breinser read early chapter drafts and Rick Chartrand's insightful opinions greatly improved some of the later scctions. The list of students who havc suffcrcd through the long cvolution of this book is now too long to present, but I would like to mention Brooke Sar gent, whose meticulous class notes were the basis of the first draft, and esse Johnson, who has worked tirelessly to improve the presentation of the many exercises in the book. The production team at Springer has been absolutely first-rate. My sincere thanks goes to all of them with a special nod to Sheldon P reface Axler for encouragement and advice surely exceeding any thing in his usual job description In a recent rereading of the completed text, I was struck by how frequently I resort to historical context to motivate an idea. This was not a conscious goal I set for Myself. Instead, I feel it is a reflection of a very encouraging trend ill mathematical pedagogy to humanize our subject with its history. From my own cxpcricncc, a good dcal of the credit for this movemcnt in analysis should go to two books: A Radical Approach to Real Analysis, by David Bressoud, and Analysis by Its Hi story, by F. Hairer and G. Wanner. Bressoud,s book was particularly influential to the presentation of Fourier series in the last chapter Either of these would make an excellent supplementary resource for this course While i do my best to cite their historical origins when it seems illuminating or especially important, the present form of many of the theorems presented here bclongs to the common folklore of the subjcct, and I have not attempted carcfu attributiOn. One exception is the Inaterial on the previously IlleltiOned gell era.lized Riemann integral, due independently to J aroslav Kurzweil and Ralph Henstock. Section 8. 1 closely follows the treatment laid out in Robert BartIe's rticle“ Return to the i the auth italicized plea for teachers of mathematics to supplant Lebesgue's ubiquitous integral with the generalized Riemann integral. I hope that Professor Bartle will see its inclusion here as an inspired response to that request On a personal note, I welcome comments of any nature, and I will happily share any enlightening remarks and any corrections via a link on my web pagc. The publication of this book comes ncarly four ycars after thc idca was first hatched. The long road to this point has required the steady support of many people but most notably that of my incredible wife, Katy. amid the Hurry of difficult decisions and hard work that go into a project of this size, the opportunity to dedicate this book to her comes as a pure and easy pleasure Middlebury, Vermont Stephen abbott August 2000

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Understanding Analysis【Stephen Abbott 】

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