MATH 509 Final: MATH 509 USC 509S05Final Solutions

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31 Jan 2019
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The subject of these notes is one of my favorites in all mathematics, and it"s not hard to say why. As you will see, the basic theorem, the banach contraction principle, has a simple and elegant statement, and a proof to match. And yet, at the same time it is extremely powerful, having as easy consequences two of the most important foundations of advanced analysis, the implicit function theorem and the local existence and uniqueness. But there is another aspect that i nd very appealing, and that is that the basic technique that goes into the contraction principle, namely iteration of a mapping, leads to remarkably simple and e ective algorithms for solving equations. But in case you want to know, the next few paragraphs are a short course on metric. Spaces that will tell you all you need to know for current purposes.

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