Lecture 1 (Dutta): Math Preliminaries

Original article (link) posted: 13/09/2005

The class by professor Dutta started with mathematical preliminaries. It might be good to study those concepts and some theorem again because I left myself unclear of some of them during taking a math class in my first year.
Well, I am thinking to write a brief summary for each week (The class is once every week on Monday). Here is the first one.

Topics in the class

1) Correspondings and Maximum Theorem
2) Contraction Mapping Theorem

What we covered in the class

In (1):
(a) The definitions of correspondings and several versions of continuities: upper semi-continuity (USC), lower semi-continuity (LSC) (and continuity).
(b) Maximum Theorem and its proof. Note) Maximum Theorem says that the maximum value is continuous and the maximizer is USC in parameters under some conditions.

In (2):
(c) The def. of contraction, Cauchy sequences and complete metric space.
(d) Contraction Mapping Theorem and its proof. Note) The theorem says that if there is a contraction corresponding and its domain is a complete metric space, then there exists a unique fixed point.


(a) I've often mixed up USC and LSC, but finally the difference seems to be clear for me.
(b) I need to reconsider the proof. It's not so complicated but not that easy either.
(c) I realized that I had forgotten the def. of complete metric space...
(d) The proof is much easier than (b). Uniqueness is almost straight forward.

Recommended readings

SLP Chapter 3
Sundaram (1995) "A Course in Optimization Theory"

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