18.100BC Analysis I -- Fall 2008

IMPORTANT INFORMATION: Solutions for Final Exam. Class Average 78 - well done!
Scores for the final are on stellar; final letter grades will be on websis by Thursday .... happy holidays!

Exams of 100B MWF and 100C can be picked up Wednesday 9:30-11:30 and 3:00-4:00 in 2-246.

The graph of the Weierstrass function:

### Course Information:

Instructor: E-mail: Lectures: Office hours: Gigliola Staffilani (B) Todd Kemp (B) Katrin Wehrheim (C) gigliola(at)math.mit.edu tkemp(at)math.mit.edu katrin(at)math.mit.edu MWF 12-1 in 4-237 TR 1-2:30 in 4-163 MWF 11-12 in 2-102, extra C session Tue 2:30-4:30pm, Thu 3-5pm in 2-246 Thu 10-12am in 2-175 Thu 12/11 4pm, Fri 12/12 11am-12:30 in 2-277 Course assistants: Tim Nguyen (B and C) Joel Lewis (C) Susan Ruff (C) Email: timothyn(at)math.mit.edu jblewis(at)math.mit.edu ruff(at)mit.edu Tue 1-2pm, Fri 1:15-2:15pm in 2-310 Fri 2-4pm in 2-333

Homework and Practice Problems: (If you are stuck on some problem consider looking at the extensive solution writeups from the C section.)
• Practice Problems from week 1 in 18.100C
• Problem Set 1, covering week 1 and 2, due 9/17. Solutions
• Problem Set 2 and notes on sequential compactness, covering week 3, due 9/24. Solutions
• Problem Set 3, covering week 4, due 10/1. Solutions
• Problem Set 4, covering week 5, will not be graded due to Midterm that week.
(You may however submit solutions as usual to get feedback.) Solutions
• Practice Problems for Midterm 1. Solutions
• Problem Set 5, covering week 6, due 10/15. Solutions
• Problem Set 6, covering week 7, due 10/22. Solutions
• Problem Set 7, covering week 8, due 10/29. Solutions
• Problem Set 8, covering week 9, due 11/5. Solutions
• Problem Set 9, covering week 10, will not be graded due to Midterm that week.
(You may however submit solutions as usual to get feedback.) Solutions
• Practice Problems for Midterm 2. Solutions
• Problem Set 10, covering week 11, due 11/19. Solutions
• Problem Set 11, covering week 12, due 11/26. Solutions
• Problem Set 12, covering week 13 and 14, due Friday 12/5. Solutions
• Practice Exam for the final. Solutions

• Tentative Syllabus: The following is a rough week-by-week guide to what we plan to cover and when.
• Week 1 (9/3,5 resp. 9/4): Ordered sets, fields, the real numbers, countability
• Week 2 (9/8,10,12 resp. 9/9,11): euclidean spaces, metric spaces
• Week 3 (9/15,17,19 resp. 9/16,18): compact sets, connected sets
Read Rudin sections 2.30-47 and notes on sequential compactness.
• Optional: additional notes on connected components.
• Week 4 (9/24,26 resp. 9/23,25): convergence
• Week 5 (9/29,10/1,10/3 resp. 9/30,10/2 ): completeness, construction of the real numbers
Read Rudin sections 3.7-14 and notes on construction of the reals.
• Monday 10/6, 4-6pm in 32-124 : Optional Review Session for Midterm 1 by Tim Nguyen
• Tuesday 10/7, 7:30-9:00pm in 4-370: Midterm 1, covering week 1-5 : exam and solutions. Scores are also on stellar.
• Week 6 (10/6,8,10 resp. 10/9,10/11): Sequences and series.
Read Rudin pgs. 3.20-3.55 and notes on conditional convergence .
• Week 7 (10/15,17 resp. 10/14,16): l^p spaces, continuity.
Read Rudin sections 4.1-12; additional material in lectures on l^p spaces, and optional notes on completeness.
• Week 8 (10/20,22,24 resp. 10/21,25): Continuity and compactness.
• Week 9 (10/27,29,31 resp. 10/28,30): Differentiability, Mean value theorem.
Read Rudin sections 5.1-15, and optional notes on the chain rule.
• Week 10 (11/3,5,7 resp. 11/4,6): Taylor series, Riemann-Stieltjes integral.
• Wednesday 11/12, 4-6pm in 4-163 : Optional Review Session for Midterm 2 by Tim Nguyen
• Thursday 11/13, 7:30-9:00pm in 1-190: Midterm 2, covering week 5-10 : exam and solutions. Scores are also on stellar.
• Week 11 (11/12,14 resp. 11/13): Integrability.
Read Rudin sections 6.12-17. and notes on Riemann integrability and continuity almost everywhere.
• Week 12 (11/17,19,21 resp. 11/18,20): Fundamental theorem of calculus, Sequences of functions.
• Week 13 (11/24,11/26 resp. 11/25): Uniform convergence.
Read Rudin sections 7.10-17 and notes on Devil's Staircase function.
• Week 14 (12/1,3,5 resp. 12/2,4): Uniform convergence, equicontinuity.
Read Rudin sections 7.18-27 and optional notes on Weierstrass function.
• Week 15 (12/8,10 resp. 12/9): Final review.
• December 12, 2-4pm in 2-105 Optional Review Session by Joel Lewis
• December 15, 1:30 - 4:30PM in Johnson: Final Exam. Solutions.
• December 16: Scores for the final are on stellar; final letter grades cannot be displayed due to MIT policy .... happy holidays!

• Literature:
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, third edition.