A metric space is a set of elements and and a metric

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Unformatted text preview: ic satisfies: iff (positivity). (symmetry) (triangle inequality). Example. A normed linear space is a metric space with same thing as a vector space.) ■ A sequence converges to if as . (A linear space is the . If , then convergence corresponds to uniform convergence of a sequence of functions. A sequence is Cauchy if, an integer such that . From Analysis, every convergent sequence is Cauchy. However, not every Cauchy sequence converges. Example. The set of numbers with . The sequence is Cauchy but does not converge (in the sense that its limit does not belong to ). ■ A metric space is complete if every Cauchy sequence converges to an element of . A complete normed linear space is a Banach space. Contraction Maps Meiss uses map as a synonym for operator. Let be a map on a complete metric space such that for all . is a contraction if there is a constant . Remark. The following little exercise in geometric series is used in the following theorem. Let integers and . , 5 Chapter 3 , subtract , . Theorem 3.4 (Contraction Mapping). Let space . Then has a unique fixed point be a contraction on a complete metric . Proof. As in the text. Plan to give it. Example (source: http://www.math.uconn.edu/~kconrad/blurbs/analysis/contraction.pdf ) Consider the complete metric space theorem applied to leads to for and some between with . The mean value and . We then have . Since for , is a contraction. Starting from any , repeated applications of gives a sequence that converges to a unique fixed point. Test this by entering on your calculator and repeatedly pressing the cosine key. The unique fixed point ■ Lipschitz Functions Let . A function is Lipschitz if for all there is a constant such that . The smallest such constant is the Lipschitz constant for in . Lemma 3.5. A Lipschitz function is uniformly continuous. Proof. is uniformly continuous means . But Lipschitz implies continuous with .□ The following example shows that such that . Therefore is uniformly need not be differentiable in order to be Lipschitz. 6 Chapter 3 . sketch Define piecewise by Example. . For . The same statement holds if zero. For constant =1. ■ and , A function Lipschitz on . is locally Lipschitz if for all Example. is not locally Lipschitz at sufficiently small that are both less than or equal to . is Lipschitz with there is a neighborhood . For any there is is with .■ , implying Lemma 3.6 Let be a function on a compact, convex set . Then condition on with Lipschitz constant . Proof. For and . Since such that satisfies a Lipschitz let . parameterizes the straight line between is convex, this line is within . sketch A, x, y, line By the Fundamental Theorem of Calculus and the chain rule: Since is compact and is continuous, has a maximum value on . Thus .□ Corollary 3.7 If is on an open set , then it is locally Lipschitz on . Proof. For each there is a closed bal...
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