Analysis II Lecture 17 Part 2 contraction mapping theorem I

We state and prove the first form of the contraction mapping theorem for complete metric spaces. The proof illustrates several useful techniques that appear in many contexts. One of them is repeated use of the triangle inequality. Another technique is adding in sums of differences to interpolate from one point to another.

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.

[iliTheFallen]

Mr. Parzygnat, at around 10:50 and later on, should not it be m > -log_{\alpha}(…). I mean there has to be a minus sign in front of log function?

[xinyulai]

write in the opposite direction?! how did you do this?