**Analysis in R^n**

**Math 20300 Section 33**

**Winter 2018**

**Announcements**

** **- There will be no class and no office hours on Friday, February 9th (College break).

**General Information**

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

E-mail: chengdr "at" uchicago.edu

Office: Eckhart 409

Office hours: Tue 5pm - 6pm at Eckhart 117

Fri 2:00pm - 3:30pm at Eckhart 409

Class hours: MWF 10:30 - 11:20am - Pick 016

Textbook: *A Textbook for Advanced Calculus, *Boller & Sally

Exam dates: There will be two in-class midterms and a final exam.

1st midterm on Friday, January 26th

2nd midterm on Monday, February 19th

Final exam on Monday, March 12th, 10:30am - 12:30pm

Grading:

Homework 10%

1st midterm 25%

2nd midterm 25%

Final 40%

**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.

Homework 1 (due Jan 10) [PDF]

Homework 2 (due Jan 17) [PDF]

Homework 3 (due Jan 24) [PDF]

Homework 4 (due Jan 31) [PDF]

Homework 5 (due Feb 7) [PDF]

Homework 6 (due Feb 14) [PDF]

Homework 7 (due Feb 21) [PDF]

Homework 8 (due Feb 28) [PDF]

**Topics covered **

This part will be updated each week to record the topics actually covered in each lecture. Some lectures are accompanied by supplementary notes.

**Week 1**

1/3 Properties of Q; brief review of Cauchy sequences; constructing R out of Cauchy sequences in Q. [Notes]

1/5 Field structure of R. [Notes]

**Week 2**

1/8 Total ordering on R; compatiblity with field structure. [Notes]

1/10 Least upper bound property of R. [Notes]

1/12 Increasing/decreasing sequences; limsup and liminf; the Cauchy criterion; subsequences.

**Week 3**

1/15 MLK day.

1/17 More on subsequences; Bolzano-Weierstrass theorem.

1/19 Convergence in R^d; definition and basic examples of metric spaces.

** Week 4**

1/22 Examples of metric spaces; open sets. [Notes]

1/24 More on open sets; closed sets; the closure. [Notes]

1/26 Midterm 1. [Solutions]

**Week 5**

1/29 The closure and the interior; examples.

1/31 Accumulation points; isolated point; relative openness and closedness. [Notes]

2/2 relative openness and closedness (continued); completeness; examples of complete metric spaces. [Notes]

** Week 6**

2/5 Continuous functions between metric spaces; Lipschitz functions.

2/7 Basic properties of continuous functions; completeness of the space of bounded continuous functions. [Notes]

2/9 College break.

** Week 7**

2/12 More on continuous functions; Uniform continuity. [Notes]

2/14 Examples and properties of uniformly continuous functions. [Notes]

2/16 Open coverings; Definition and basic examples of compact sets; Some consequences of compactness. [Notes]

** Week 8**

2/19 Midterm 2. [Solutions]

2/21 Continuous functions on compact sets. [Notes]

2/23 Continuous functions on compact sets (continued); compactness in Rd (with the usual metric). [Notes]