Analysis II Lecture 17 Part 4 weaker fixed point theorem for compact subsets скачать в хорошем качестве

Analysis II Lecture 17 Part 4 weaker fixed point theorem for compact subsets

Equivalent definitions for what it means for a subset of a metric space to be compact are given. Although the open cover definition agrees with the sequential definition (any sequence in it has a convergent subsequence), compact subsets are not always closed and bounded. Nevertheless, there is a version of the contraction mapping theorem for compact subsets of metric spaces assuming that the function is only distance decreasing (which is weaker than the function being a contraction). This is part of a series of lectures on Mathematical Analysis II. Topics covered include continuous and differentiable multi-variable functions on Euclidean space, the chain rule, the implicit function theorem, manifolds, tangent spaces, vector fields, the degree and index of a smooth map, the Euler characteristic, metric spaces, the contraction mapping theorem, existence and uniqueness of solutions to ordinary differential equations, and integral equations. I speak rather slowly, so you may wish to increase the speed of this video. These videos were created during the 2017 Spring semester at the UConn CETL Lightboard Room.