**Analysis in Rn I (accelerated)**

**Math 20310 Sections 33 & 41**

**Autumn 2018**

**Announcements**

- Some of the information here depends on which section you are in. Please make sure you have your correct section number.

- First-day handout: [Section 33][Section 41]

**General Information**

** **Instructor: Da Rong (Daren) Cheng

E-mail: chengdr "at" uchicago.edu

Office: Eckhart 409

Office hours: TF 2pm - 3:30pm

College Fellow: Ishan Banerjee

E-mail: ishan "at" math.uchicago.edu

Office: Eckhart 127

Office hours: M 2pm - 3:30pm Th 11am - 12:30pm

Problem session: M 6pm - 7pm at Eckhart 202

Textbook: "*Principles of Mathematical Analysis (3rd ed.)" *by Walter Rudin. (Chapters 1 - 4.)

Class hours:

Section 33: MWF 10:30am - 11:20am at Eckhart 308

Section 41: MWF 11:30am - 12:20pm at Eckhart 203

Grading:

Homework 10%

1st midterm 25%

2nd midterm 25%

Final 40%

**Exam Dates**

There will be two in-class midterms and one final exam. The midterm dates are the same for both sections; HOWEVER the final exam dates are different.

Midterms:

1st midterm on Friday, October 26th, in class (Week 4)

2nd midterm on Friday, November 16th, in class (Week 7)

Final exam:

Section 33: Monday, December 10th, 10:30am - 12:30pm at TBD

Section 41: Wednesday, December 12th, 10:30am - 12:30pm at TBD

**Homework **

Homework will generally be assigned every Wednesday and due the following Wednesday in class. Late homework will not be accepted under all circumstances. However, at the end of the quarter, the lowest homework score will be dropped. (This means that you can miss one homework without penalty.) Solutions to selected problems will be posted after each homework is due.

Homework 1 (due 10/10) [PDF][Solutions]

Homework 2 (due 10/17) [PDF][Solutions]

Homework 3 (due 10/24) [PDF]

**Topics covered **

This part will be updated each week to record the topics actually covered in each lecture.

**Week 1**

10/1 Overview of course; motivation for the construction of R.

10/3 R as Dedekind cuts; ordering on R.

10/5 The least upper bound property; addition and multiplication on R.

**Week 2**

10/8 Addition and multiplication on R (continued); embedding Q in R; an application of the l.u.b. property. [Notes on multiplication]

10/10 Finiteness, countability and uncountability; examples and basic properties. [Details of the last proof]

10/12 Countable union of countable sets; Cantor's diagonal process; Definition of metric spaces. [Corollary on page 29 of Rudin]

** Week 3**

10/15 Young, H\"older and Minkowski inequalities; interior points and limit points.

10/17 The interior and the derived set; open sets and closed sets; relation between the two concepts. [Theorem 2.20]

10/19 Closure; density and separability; relative open- and closed-ness.

** Week 4 (tentative)**

10/22 Compact sets; basic consequences of the definition; examples and counterexamples. (p.36 - 38)

10/24 Compactness in Rk. (p. 38 - 40)

10/26 Midterm 1.