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A note to the reader

Few people rely solely on any social science for their pleasures, and attaining a suitable level of ecstasy involves work. . . It is a nuisance, but God has chosen to give the easy problems to the physicists.

Lave and March (1975)

Some people read mathematics books for pleasure. I assume that you are not one of this breed, but are studying this book to enhance your understanding of economics. While I hope this process will be enjoyable, to make the most of it will require some effort on your part. Your reward will be a comprehension of the foundations of mathematical economics; you will appreciate the elegant interplay between economic and mathematical ideas; you will know why as well as how to use particular tools and techniques.

One of the most important requirements for understanding mathematics is to build up an appropriate mental framework or structure to relate and integrate the various components and pieces of information. I have endeavored to portray a suitable framework in the structure of this book, in the way it is divided into chapters, sections and so on. This is especially true of the early mathematical chapters, whose structure is illustrated in the following table:

Sets Functions
Ordered sets Monotone functions
Metric spaces Continuous functions
Linear spaces Linear functions
     Convex sets       Convex functions
     Cones       Homogeneous functions

Please keep the framework in mind as you proceed through the book.

You will also observe that there is a hierarchy of results. The most important results are stated as theorems. You need to be become familiar with these, their assumptions and their applications. Important but more specialized results are stated as propositions. Most of the results, however, are given as exercises. Consequently, exercise has a slightly different meaning here than in many texts. Most of the exercises in the book are not "finger exercises", but substantive propositions forming an integral part of the text. Similarly, examples contain many of the key ideas and warrant careful attention.

There are two reasons for this structure. First, the exercises and examples break up the text, highlighting important ideas. Second, the exercises provide the potential for deeper learning. It is an unfortunate fact of life that, for most of us, mathematical skills (like physical skills) cannot be obtained by osmosis through reading and listening. They have to be acquired through practice. You will learn a great deal by attempting to do these exercises. In many cases, elaborate hints or outlines are given, leaving you to fill in the detail. Then you can check your understanding by consulting the comprehensive answers.