Transcription of Chapter 3 The Contraction Mapping Principle
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Chapter 3 The Contraction Mapping Principle
Chapter 3 The Contraction Mapping PrincipleThe notion of a complete space is introduced in Section 1 where it is shown that everymetric space can be enlarged to a complete one without further condition. The importanceof complete metric spaces partly lies on the Contraction Mapping Principle , which isproved in Section 2. Two major applications of the Contraction Mapping Principle aresubsequently given, first a proof of the Inverse Function Theorem in Section 3 and, second,a proof of the fundamental existence and uniqueness theorem for the initial value problemof differential equations in Section Complete Metric SpaceInRna basic property is that every Cauchy sequence converges. This property is calledthe completeness of the Euclidean space.
3.2. THE CONTRACTION MAPPING PRINCIPLE 3 This theorem will be further explained and proved in the appendix. 3.2 The Contraction Mapping Principle Solving an equation f(x) = 0;where f is a function from Rn to itself frequently comes up in application. This problem can be turned into a problem for xed points. Literally,
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