10-327/Classnotes for Monday December 6
See some blackboard shots at.
|Dror's notes above / Student's notes below|
- Question. The fact that the metric space of real-valued functions on the unit interval with uniform metric is complete uses the fact that [0,1] is compact right? If the function space is defined on a non-compact topological space is that necessarily complete?... -Kai Xwbdsb 00:01, 20 December 2010 (EST)
- No, the compactness of [0,1] is not used. As we said in class, if (fn) is Cauchy in the uniform metric, then for any x, the sequence (fn(x)) is Cauchy in , so it has a limit. Call that limit f(x); it is not hard to show that f is continuous and that . Theorem 43.6 in Munkres is a slight generalization of this. Drorbn 07:12, 20 December 2010 (EST)
Everybody good luck on the exam!-Kai
Great course! Thank you very much for all your help Dror and all the classmates in this class. -Kai