Instructor: Mladen Bestvina
Office: JWB 210
Office hours: By appointment
Meets: MWF 2-2:50 in LCB 225
Midterm: Friday, Oct. 6
Final: Tuesday, December 12, 2023, 1:00 – 3:00
pm
Text: There are many textbooks on
graduate real analysis and in some sense a lot of them are
roughly equivalent (the first Tao's book covers measure theory
and the second covers Banach spaces). I will loosely follow
but here are some others that you may find useful:
Homework: It
will be assigned weekly. You are encouraged to work in groups,
but what you write should be your own work and you should list
the other people in your group. It will be due every week on
Mondays at 9 am and you should turn it in through canvas in
latex. Late homework is not accepted but the lowest
two scores are dropped from the count. You should read the
assigned reading for the week before the corresponding
lecture. I will also typically give out several problems each
week that you are not required to turn in.
Problem sessions: We'll
have a problem session every other week outside the class
time where you are expected to present solutions to
unassigned problems. These will also act as group office
hours. It will be in JWB 333 on Fridays at 3.
Grading: The final
grade is based on homework (30%), problem session activity
(10%), the midterm (20%) and the final (40%).
| Week |
Reading |
Homework |
Due Date |
| 1 |
Tao 1.1-1.2, Axler Ch. 1: Jordan content,
Riemann-Darboux integral, measure |
1,5,7,9,15 from hw01 |
8/28 |
| 2 |
Folland Ch. 1 or Tao, about the first half of
1.2 |
1,2,5,6,7 from hw02 |
9/5 |
| 3 |
Properties of Lebesgue measure, Measurable
functions and integration |
1,3,4,5,10 from hw03 |
9/11 |
| 4 |
Monotone limit theorem, Fatou's lemma,
Lebesgue dominated convergence theorem |
2,3,7,9,10 from hw04 |
9/18 |
| 5 |
Modes of convergence, Product measures,
Fubini-Tonelli, Monotone class theorem |
1,3,4,5,8 from hw05 |
9/25 |
| 6 |
A bit more on Lebesgue measure. Signed
measures. Hahn and Jordan decomposition theorems. |
2,5,7,8,9 from hw06 |
10/2 |
| 7 |
prepare for the midterm on Oct 6. No homework
this week. |
||
| 8 |
Lebesgue decomposition and Radon-Nikodym
derivative |
2,4,5,7,9 from hw07 |
10/23 |
| 9 |
Lebesgue differentiation, monotone functions
and BV functions |
1,2,3,6,9 from hw08 |
10/30 |
| 10 |
Banach spaces, L^p spaces, Hahn-Banach |
3,4,5,8,9 from hw09 |
11/6 |
| 11 |
Baire category, Open Mapping, Closed Graph,
Banach-Steinhaus |
3,5,7,8,12 from hw10 |
11/13 |
| 12 |
Riesz representation, weak and weak* topology |
2,3,6,8,10 from hw11 |
11/20 |
| 13 |
Hilbert spaces |
4,7,9 from hw12 |
11/27 |
| 14 |
Stone-Weierstrass, Fourier transform |
1,2,4,5,8 from hw13 |
12/4 |
| 15 |
Fourier transform. I'll follow the following
concise notes http://www.math.uchicago.edu/~womp/2006/Fourier.pdf |
hw14 |
for your enjoyment |
You can contact me by email.
Required
statements:
Safety on campus