xvi Preface Read every word. That means pronounce it in your head. This is the opposite of what you should do when you read a novel, but it is the only way I know to learn reading mathematics. In particular I frequently sprinkle throughout the text the parenthetical remarks “Why?” and “Verify.” That means I think it important that you heed that question/command. You will frequently see the phrase, “the details are left to the reader.” Supply those details. I do not do this because I am lazy but I think what is required to fill in the details is routine and within your reach. You can think of it as a speed bump in your reading or as a test of your comprehension. If you cannot complete the argument, you have missed something and should put on the brakes and back up to the previous result. I certainly want to give you a view of the whole forest, but I also want you to become familiar with the individual trees right down to the patterns of the bark. You will also see many examples. I have said on more than one occasion that “Mathematics is a collection of examples.” Without examples, the theory is vacuous. In fact, the way mathematical concepts have come to be is the observation that several examples and arguments have a commonality and that it is worthwhile to single out the essential ingredients. Results are statements about a collection of examples. Then there are the exercises. They run the gamut from the routine to the challenging. Spending time on exercises builds your ability to understand and do mathematics. If you are having trouble solving a particular exercise, follow the standard advice: add a reasonable additional hypothesis and see if you can then solve it. Try to construct a counterexample to the exercise and see why it cannot be done. Using Wikopedia is not frowned on, certainly not by me. In fact, you will see some references in the text to Wikopedia and other web sites. Hopefully they will still exist when the book hits the street. The mathematics topics on the web are frequently well done but sometimes too shallow for you. So if you come across some topic that is a little fuzzy for you, go there and see what it says. It may help or not. If not, try going to your teacher. Finally there are a few historical notes. I think they are interesting and hope you do too. On some topics I found the history opaque. For example the development of the Spectral Theorem for hermitian matrices is a historical mystery to me and maybe deserves a scholar’s effort. Again feel free to browse the web for additional historical notes, though like all that is there, you might want to cross check it with a source that has been subjected to peer review — like a book. ***** I want to thank several people who have helped me get this ready for publication. Three of my undergraduates at George Washington Univer- sity, Diane Holcomb, Sam Mendelson, and Katie Walsh, worked through

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.