MAST30026 Lecture 13: Metrics on function spaces (Part 2)

I discussed pointwise and uniform convergence of functions, proved that the uniform limit of continuous functions is continuous, and used that to prove that Cts(X,Y) is a complete metric space with respect to the sup metric if X is compact and Y is a complete metric space.

Have questions? I hold free public online office hours for this class, every week, all year. Drop in and say Hi! For details see the class webpage.

[friedrichschumann]

What is going on with video quality here? It seems, the video had been compressed too much.

Amazing: Between 22:07 and 22:12 the video quality increases dramatically. From there on the text on the blackboard can be read again.

[friedrichschumann]

45:35 f_n(x) = e, not 1. I.e. also lim_{n\to\infty} f_n(x) = e * \delta_0, not just \delta_0.
It doesn't affect the point made, though.

Edit: Just realizing lim_{x\to 1/n -} exp(1/(1-n^2 x^2)) = \infty.
A minus sign in the expression will do the job.