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Elementary Real Analysis: Fall 2001 TEXT: Michael Spivak, Calculus, 3rd edition, Houston: Publish or Perish, Inc., 1994 Four main goals you should have for this term course: 1. To learn basic - d techniques of proving theorems in analysis, and to master the basic consequences of the Least Upper Bound Axiom. 2. To become conversant with a classical body of knowledge, including pertinent examples and counterexamples, about limits, continuity, derivatives, and integrals, at the "advanced calculus" level. 3. To develop skill in understanding more sophisticated proofs in analysis. 4. To understand the theory of infinite series, and to gain some sense of what makes a series converge of diverge, and at what rate. Many standard and important topics in real analysis will hardly be mentioned. Topological issues will receive scant attention and the development of real analysis in the more general setting of metric spaces must await another course. The theory of the integral will receive less attention than it deserves in order to allot more time to the study of sequences and series. In the syllabus, you will notice that some chapters should be studied carefully and others merely skimmed before you attack the problem sets. Unlike the authors of many other introductory analysis texts, Spivak devotes many paragraphs to motivate definitions and theorems and to give readers some sense of how modern mathematicians view their subject. Consequently, the more diligently you read the text, the more you will get out of the course. Solutions to problems should be written carefully (and legibly) and include a reasonable amount of detail. If challenged, you should certainly always be able to supply missing steps. Some of the problems are quite challenging--include your ideas about problems you could not completely solve, if you think there is some substance to these ideas. Spivak's Calculus contains several appendices and sections that are worth reading, although they will not be assigned. Chapter 16 contains a mysterious proof that pi is irrational, and chapter 21 a proof that e is transcendental. This chapter also includes some facts about Cantor's theory of countable and uncountable sets, including the important theorem that, on an interval, monotonic functions can have at most a countably infinite number of discontinuities. Chapter 25 contains a rigorous introduction to complex numbers. The epilogue includes a construction of the real numbers and a proof of their uniqueness as a complete ordered field. (My favorite method of constructing the real numbers, for example, is not the same as Spivak's. But we get the same complete ordered field of real numbers in the end, anyway, although his real numbers are sets of rational numbers and mine are equivalence classes of Cauchy sequences of rational numbers.) In

chapter 26, you will find an elementary proof of the Fundamental Theorem of Algebra. For those who like physical applications, Kepler's Laws are treated in chapter 17. There are many good textbooks on advanced calculus and real analysis. Listed below is a sample that I am familiar with, along with several texts that might motivate you to learn more analysis. Old-fashioned courses in advanced calculus emphasized techniques (of integration, of summing series, of partial differentiation with the chain rule, etc.), special functions (e.g., the gamma function), special classes of numbers (e.g. Bernoulli numbers), and introductions to topics like Fourier analysis and Laplace transforms, which might be useful in the student's physics and engineering courses. Though there is no universal standard, the more modern courses tend to emphasize the theory of the calculus, connections with linear algebra via vector calculus, topological concepts, and axiomatic development of the subject, with attention to algebraic structure. With the exception of Hardy's and Widder's texts, the books below follow a more modern approach overall. · Lang, Serge, Undergraduate Analysis, New York: Springer-Verlag, 1983 Lang has written numerous textbooks. I have profited most from his three analysis texts. (The other two are devoted to graduate level real and complex analysis.) This is a fairly comprehensive, nicely organized, and up-to-date treatment of the subject, with some very interesting problems. Since most of Lang's many undergraduate textbooks are written in the same style, with the same terminology and notation, you can easily dip into others, once you have perused one of them. · Rudin, Walter, Principles of Mathematical Analysis, New York: McGraw-Hill Book Co., 1964 Considered by many to be the classic introductory text in real analysis in English, Rudin's text first appeared in 1953. The Cambridge philosopher of science, Imre Lakatos, referred to its "infallabilist style" and, while criticizing the philosophy behind such a style, praised it as one of the best in the "Euclidean tradition." In Rudin, all is clear, crisp, and concise, but you will not find the leisurely motivational essays that appear so frequently in Spivak. · Bartle, Robert, The Elements of Real Analysis, John Wiley & Sons, Inc., 1964 This text covers the same material as Rudin's at the same level. I have always found it more enjoyable reading than Rudin. Bartle has more extensive problem sets and provides some reasons for why one might want to study this material. Bartle has written a more recent textbook on analysis; in this new text, he develops the theory of the Henstock integral, which appears to be a natural generalization of the Riemann integral, while lacking the well-known defects that the Lebesgue integral corrected. The increasing number of articles advocating the Henstock integral suggests that undergraduate math majors may soon be exposed to this routinely.

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Hardy, G. H., A Course of Pure Mathematics, 10th edition, Cambridge University Press, 1952 The first edition appeared in 1908. In 1992, the Cambridge Mathematical Library reissued this classic text in paperback. Both the sections and the problem sets are at times almost breathtaking. It is beautifully written, with a proselytizer's enthusiasm for the subject. Many of the problems come from the Cambridge Mathematical Tripos examinations. A good book for a desert island--you will never be bored. And you can always ponder the photograph of Hardy on the front cover. · Abbot, Stephen, Understanding Analysis, New York: Springer-Verlag, 2001 One of the more recent additions to the literature, Abbot's text has received wonderful reviews. Abbot's basic contention is that the subject can seem dull and unmotivated to the typical undergraduate. Therefore, he motivates the subject with discussions introducing each chapter. Rather than focus of the proofs of "obvious" theorems, he emphasizes the non-intuitive results that clearly demand careful proof. The text also contains sections that are really extended projects. · Goldberg, Richard, Methods of Real Analysis, 2nd edition, New York: John Wiley & Sons, Inc, 1976 Goldberg develops the theory in the more general setting of metric spaces. The first third of the book carefully introduces the topological concepts critical to a thorough understanding of the subject. The proofs are so thorough that this book is useful for self study. The problem sets test knowledge of the material presented, but otherwise tend not to be very challenging or exciting. Widder, David, Advanced Calculus, 2nd edition, New York: Dover Publications, 1989 This text was first published in 1947; the second edition appeared in 1961. The author was a professor at Harvard. It contains a lot of interesting mathematics, mostly multivariable calculus. At first glance, much of the book looks like an advanced first year calculus text, but the theory is developed carefully, and the theorems are proved. The problem sets contain interesting and sometimes very challenging problems. A nice place to get a quick introduction to Stieltjes integrals, the gamma function, Fourier series, and the Laplace transform. Corwin, L. J. and Szczarba, R. H., Calculus in Vector Spaces, 2nd edition, New York: Marcel Dekker, Inc. 1995 Although Marcel Dekker publishes perhaps the ugliest textbooks on the market, this book by Corwin and Szczarba is worth owning and studying. In one book, you can learn linear algebra, multivariable calculus, elementary analysis, and topology. The authors accomplish this by banishing all the neat and interesting classical mathematics found in Widder's text and focusing on material that lends itself well to a rigorous axiomatic treatment. Here is the start of a typical (and very instructive) section: "One of the more useful notions in the calculus of one variable is the higher derivative. We now develop the analogous notion for functions between inner product spaces." It is left to the reader · ·

to determine why one cannot get higher derivatives more directly, without considering inner product spaces. · Barnsley, Michael, Fractals Everywhere, Boston: Academic Press, Inc., 1988 In this book, Barnsley makes the abstract concepts of elementary real analysis and topology look utterly natural and obvious. If you want to learn some attractive geometric applications of real analysis and see how theorems are really used to deduce important results, this is a great book to read. There is a second edition. · DeBruijn, N.G., Asymptotic Methods in Analysis, 2nd edition, New York: Dover Publications, Inc., 1981 This is one of my favorite books. DeBruijn develops some theory here, but his main interest is in demonstrating, through a series of important examples, how powerful the theory is. He attacks difficult problems and works out their solutions in detail, providing a model for students and researchers. He devotes ten pages, for example, to a treatment of the iteration of the sine function (which has a neutral fixed point at 0 and therefore resists easy analysis). Ideas from analysis are constantly required to understand the concepts and prove the theorems in this book. Any of the many introductory undergraduate texts on discrete dynamical systems illustrate how important it is to have a good grasp of real analysis. Texts by Robert Devaney and Denny Gulick are good examples.

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