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Intro Real Analysis, Lec 34: p-Norm, Sup Norm, Continuity & Preimages, Images of Compact Sets
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Introduction to Real Analysis Lecture 34. Topology Unit, Part 4.
(0:00) Plan for coming lectures.
(0:20) p-Norm on R^n and corresponding metrics.
(7:01) Supremum (Uniform/Infinity/Max) Norm on C[a,b].
(14:42) Sequences and convergence of sequences in metric spaces.
(20:15) Continuity of functions between metric spaces.
(23:42) The rest of the lecture will involve stating and proving two very important theorems related to continuity, preimages of open sets, and direct images of compact sets.
(25:34) State and prove the theorem about the fact that preimages of open sets under continuous maps are open.
(44:27) State and prove the (partial) generalization of the Extreme Value Theorem that direct images of compact sets under continuous maps are compact.
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