Elementary Theory of Metric Spaces

Abstract

In Chapters 2 through 5 we developed many properties of functions from jRl into jRl with the purpose of proving the basic theorems in differential and integral calculus of one variable. The next step in analysis is the establishment of the basic facts needed in proving the theorems of calculus in two and more variables. One way would be to prove extensions of the theorems of Chapters 2-5 for functions from jR2 into jRl , then for functions from jR3 into jRl , and so forth. However, all these results can be encompassed in one general theory obtained by introducing the concept of a metric space and by considering functions defined on one metric space with range in a second metric space. In this chapter we introduce the fundamentals of this theory and in the following two chapters the results are applied to differentiation and integration in Euclidean space in any number of dimensions. We establish a simple version of the Schwarz inequality, one of the most useful inequalities in analysis. Theorem 6.1 (Schwarz inequality). Let x = (Xl' X2, "" XN) and Y = (Yl, Y2,"" YN) be elements ofjRN. Then I N I (N)1/ 2 (N)1 / 2