x PREF third feature is the real kicker in this business. We list a series of challenge problems that increase in impossibility as the chapters progress. I have taught the material in this book many times over the past forty- five years. The main audience has been the students in Honors Analy- sis (MATH 207-208-209) at the University of Chicago. These students are drawn from two sources. The first is the collection of sophomores who have excelled at Honors Calculus in their first year at Chicago. The second is a selection of pyrotechnically endowed freshmen who are capable of attacking mathematics at this level. Some of the texts I have recommended during this time are T. Apostol, Mathematical Analysis [2], J. Dieudonn´ e , Foundations of Modern Analysis [3], A. Kolmogorov and S. Fomin, Introductory Real Analysis [12], S. Lang, Undergraduate Analysis [13], L. Loomis and S. Sternberg, Advanced Cal- culus [18], W. Rudin, Principles of Mathematical Analysis [24], and, more recently, C. Pugh, Real Mathematical Analysis [22]. All of these books have some nice features. The intersection with the material of the present book is highly nontrivial. Nonetheless, I have always liked the idea of challenge problems, independent projects, and the organization of the mathematics presented here. For example, it is about time that mathematicians came to grips with Fourier analysis on p-adic fields, since it is an integral part of current-day research. At the beginning of each chapter, I have included a quote from a well- known mathematician (or group of mathematicians) that gives a certain perspective on the material in that particular chapter. We leave it to the reader to speculate as to whether this perspective is that of the author. These quotes express a variety of opinions, and I have found them to be informative and sometimes amusing. The quote of A. Zygmund at the be- ginning of Chapter 7 is particularly relevant to the mathematics in the text. FURTHER ADVICE TO THE STUDENT (If you do not care about advice, just get started with the challenge problems in Chapter 1.) It would be much better for both of us if I were sitting on a desk at the front of the class and talking to you. Nevertheless, a few words of warning are in order. First of all, you should scan the material in Appendices A and B and make sure you feel comfortable with it. Throughout the text, there are many references to these appendices. Secondly, if you find a par- ticular exercise in the text to be quite simple and the next exercise to be very difficult, that’s just the way it is. When doing mathematics, you never know when a road that seems smooth is going to have a pothole that is ten feet deep. Thirdly, if you take the challenge problems seriously, you will find that some of these problems can require looking somewhere other than Wikipedia. In that process, you can discover that there is lots of good stuff
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