Functions on metric spaces
We developed many of the basic properties of metric spaces in Chapter 6. A complete metric space was defined in Chapter 13 and we saw the importance of such spaces in the proof of the fundamental fixed point theorem (Theorem 13.2). This theorem, which has many applications, was used to prove the existence of solutions of differential equations and the Implicit function theorem. We now discuss in more detail functions whose domain is a metric space, and we prove convergence and approximation theorems which are useful throughout analysis.
KeywordsContinuous Function Convex Function Compact Subset Interior Point Cauchy Sequence
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