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.

**Comments**

(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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