# 240C Homework Assignments

Only hand in the problems with (*) after them. However, you should make sure that you are able to do all of the problems.

Warning:   Numbering is from the most current version of the notes.

1. Due Monday, April 5, 2004.

Chapter 23. 23.6*, 23.16*, 23.17, 23.18, 23.19, 23.20*, 23.21*

2. Due Friday, April 9, 2004.

Chapter 24. 24.1, 24.2*, 24.3*, 24.4*, 24.5*, 24.6, 24.10*, 24.13*

3. Due Monday, April 19, 2004.

Chapter 25. 25.1, 25.2*, 25.3*, 25.4, 25.5*, 25.7*, 25.13*, 25.14*,

Chapter 16. 16.1*, 16.2*.

4. Due Monday, April 26, 2004.

Chapter 25. 25.16*, 25.19*, 25.20*(see below), 25.24*, 25.26*, 25.29*, 25.34*, 25.35

5. Due Monday, May 3, 2004. (Note: the lecture notes have just been updated on 4/26/04.)

Chapter 28. 28.4, 28.6, 28.8*, 28.9*, 28.10*, 28.11*, 28.13*(see below), 28.15, 28.16, 28.17
Chapter 29. 29.1, 29.2*, 29.3*

6. Due Monday, May 12, 2004. (Last Assignment.)

Chapter 29:  29.7*, 29.8*, 29.9*, 29.10*, 29.11    (Numbering from Previous Version of the notes.)
Chapter 33:  33.4*, 33.8*, 33.9*

Below are the same problems (but numbered differently) form the current version of the notes.

# 240A Homework Assignments

Only hand in the problems with (*) after them. However, you should make sure that you are able to do all of the problems.

Warning: For old assignments, the numbers below in the HTML file may no longer correspond to the correct problem because of changes to the notes. Sorry!!  However click  here  for a PDF file of the assigned problems with the correct numbering for the current version of the notes.

1. Homework #1 is Due Monday, October 6, 2003.
Read Pages 5-21 in Part I of the lecture notes and do the following Exercises in those notes:

2.1*, 2.2, 2.3*, 2.4*, 2.5
3.1, 3.2, 3.3*, 3.8*
4.5*, 4.6, 4.7*, 4.8*

2. Homework #2 is Due Monday, October 13, 2003.  The following problems are from
Part II: Metric and Banach Space Basics:

4.3, 4.9, 4.10* (Hint: use 4.9), 4.11*, 4.12*, 4.13, 4.14, 4.15*
6.2*, 6.3*, 6.4*, 6.7, 6.8, 6.11*, 6.12, 6.13*, 6.15* (Hint: use the dominated convergence theorem.)

3. Homework #3 is Due Wednesday, October 22, 2003.  The following problems are from
Part II: Metric and Banach Space Basics:

6.10* (definitely do this problem), 6.14
7.1*, 7.2*, 7.3*, 7.4, 7.5, 7.6*, 7.7*, 7.10*

4. Homework #4 is Due Wednesday, October 29, 2003. Friday October 31, 2003.

From
Part II: Metric and Banach Space Basics:  8.1, 8.3*,  8.6*, 8.9*, 8.10*, 8.11, 8.14, 8.16*, 8.17*
[and perhaps 8.4, 8.5* if we reach these in class].

5. Homework #5 is Due Friday, November 7, 2003  Monday, November 10, 2003.

From:
Part II: Metric and Banach Space Basics:  8.4*, 8.5*
From: Part III: Topological Spaces I.: 10.1, 10.2*, 10.3, 10.4*, 10.7*
From: Lebesgue Integration Theory Notes:  18.1*, 18.2, 18.3*, 18.4*

6. Homework #6 is Due Wednesday, November  19, 2003.
From: Part III: Topological Spaces I.: 10.5, 10.6*, 10.20*, 10.21*, 10.24*
From: Lebesgue Integration Theory Notes:  Read Proposition 18.33:  18.5, 18.6*, 18.9*, 18.10*, 18.11*, 18.12

7. Homework #7 is Due Friday December 5, 2003.
(Please use the problems from the most current version of the notes when doing the homework problems.)

From: Lebesgue Integration Theory Notes:  19.1, 19.2*, 19.3*, 19.4*, 19.5*, 19.6*, 19.7*, 19.8,  19.10*(Hint: "Fatou times two."),  19.14*, 19.16*
From: Folland p. 60: (# 2.31b,e)*.
(In part b the answer is off by a "-" sign and the sum is on k.

In part e, s=a. You may also freely use the Taylor series expansion for (1-z)^(-1/2) which is convergent for |z|<1.)

# 240B Homework Assignments

Only hand in the problems with (*) after them. However, you should make sure that you are able to do all of the problems.

Warning: For old assignments, the numbers below in the HTML file may no longer correspond to the correct problem because of changes to the notes. Sorry!!

8. Due Wednesday January 14, 2004.  The following problems are from
V: Lebesgue Integration Theory: Part I. notes:

20.2, 20.5*, 20.6, 20.7*, 20.8, 20.9*, 20.10*, 20.11, 20.15*, 20.17*

9. Due Wednesday January 21, 2004.  The following problems are from
V: Lebesgue Integration Theory: Part I. notes:

20.18*, 21.1*,   21.2, 21.5*, 21.6, 21.7*, 21.9*, 21.12**, 21.13**.
Hint for 21.1*:  use the idea in the proof of the version of DCT in Corollary 21.17 just above this problem.)
(The problems lined out will be on the next homework assignment. The "**" -- means you should definitely do these problems and hand them in.)

10. Due Wednesday January 28, 2004.

(V: Lebesgue Integration Theory: Part I. notes:)
Chapter 21. 21.2, 21.5*, 21.6, 21.7*, 21.13*
(Part III: Topological Spaces.)
Chapter 10. 10.12*, 10.13, 10.15*, 10.16*, 10.17* -- current numbering
(Chapter 10. 10.10*, 10.11, 10.13*, 10.14*, 10.15* old numbering)

11. Due Wednesday February 4, 2004.

(Part III: Topological Spaces.)
Chapter 10. 10.8*, 10.9*, 10.10*, 10.18, 10.19, 10.20*, 10.21
Chapter 11. 11.1, 11.2*, 11.3, 11.4*, 11.5*, 11.6, 11.7*, 11.8, 11.9, 11.10*, 11.11

Numbering is from the current version of the notes.

12. Due Wednesday Friday February 11 13, 2004. (Note: new due date.)

(V: Lebesgue Integration Theory: Part I. notes:)
Chapter 21. 21.3*, 21.4
Part III: Topological Spaces.)
Chapter 11. 11.12*, 11.13*, 11.14, 11.15*, 11.16*, 11.19*
Chapter 12. 12.1, 12.2*, 12.4*, 12.6, 12.7*, 12.8, 12.9

There is a typo in 11.13:
please replace  \$d_A(y) = 1\$ by  \$d(0,y) = 1.\$

13. Due Wednesday Friday February 27, 2004. (Note: new due day.)

Part III: Topological Spaces.)
Chapter 12. 12.6*, 12.7*, 12.8, 12.9 (just look at this one)
Chapter 22: Approximation and Convolution
Chapter 22. 22.1*, 22.3, 22.9*, 22.10*, 22.11*

14. Due Friday March 5, 2004.
(Note, numbers have changed in Chap. 22, please refer to the newest version.)

Chapter 14: Hilbert Space Basics
Chapter 14. 14.1*, 14.2, 14.3*, 14.4, 14.5, 14.7, 14.8*

Chapter 22: Approximation and Convolution
Chapter 22. 22.3, 22.4*, 22.8*, 22.9*, 22.10*
Chapter 23: L^2 Hilbert Space Techniques.
Chapter 23. 23.7* (Corrected version of this exercise appears below.)

15. Due Friday, March 12, 2004.

Chapter 14: Hilbert Space Basics
Chapter 14. 14.6*, 14.14*
Chapter 23: L^2 Hilbert Space Techniques.
Chapter 23. 23.1*, 23.2*, 23.3, 23.4*, 23.5*, 23.8*, 23.10*

Numbering is from the most current version of the notes.

16.

# Remarks on doing the homework.

The homework is an important part of this class. The homework is your best chance to learn the material in this course. You may consult others on the problems, but in the end you are responsible for understanding the material. I suggest that you try all the problems on your own before consulting others. Even false starts on problems will help you learn.

Here is what the TA will be looking for in your solutions.

1. The solutions must be written clearly. This includes reasonable handwriting and good English. If the TA has to struggle to read what you have written, he will not grade the problem!

2. The solutions should be complete and clear. A good rule of thumb is: if you have some doubt about your solution it is probably wrong or at best incomplete.

3. Results that you use in your proof from undergraduate analysis or from the text book or from the lecture notes should be stated clearly. Here is an example of what I am looking for:

... So we have shown that fn converges to f uniformly. Since each fn is continuous and the uniform limit of continuous functions is continuous, we know that f is continuous.

The reference to a theorem from undergraduate analysis is underlined..