Most recent assignments

The final exam for this course will take place from 8:00 A.M. to 10:00 A.M. on Wednesday, December 15, 2004 in the usual classroom.

We will skip Section 7.7.

We will skip Section 5.5 on “Convex Functions.”

The textbook for the course will be as follows:

Real Analysis, Third Edition, by H. L. Royden (Pearson Education, 1988)

Let me know if you have any questions.

Assignments

1. Section 5.1: 1, 2, 3, 4
   Due September 3, 2004
   Note: The statement of problem 3 in the text is wrong. In Part (a), the statement that you should prove is

   D₊f(c)<=D⁺f(c)<=0<=D_-f(c) <=D^-f(c)

   Here “<=” means “less than or equal to.”

2. Section 5.2: 8, 9, 11
   Due September 10, 2004
   

3. Section 5.2: 10a
   Section 5.4: 13, 18, 20a, 20b
   Due September 17, 2004
   Hint: For one of the problems, you may find it helpful to use the fact that if a is less than or equal to c and c is less than or equal to b, then P_a^b=P_a^c+P_c^b and N_a^b=N_a^c+N_c^b . You do not need to prove these two facts.

4. Section 6.1: 1, 2, 3, 4
   Due September 20, 2004

5. Section 6.2: 7
   Due September 24, 2004
   Note: Assume throughout that p and q are both finite and greater than 1.

6. Section 6.2: Do the problems in the special handout.
   Due September 24, 2004

7. Section 6.3: 10, 16
   Due September 29, 2004
   Hint: For Problem 16, use Theorem 17 on Page 92.

8. Section 6.3: 11
   Due October 1, 2004
   The due date for this problem has been extended to October 4, 2004.

9. Section 6.4: 19
   Due October 1, 2004
   The due date for this problem has been extended to October 4, 2004.

10. Section 6.5: 21
    Due: October 13, 2004
    Note 1: For Part (b) of Problem 21, the solution “f(x)=0 for all x” is not acceptable.
    Note 2: The hint given for Part (b) of Problem 21 may not be completely correct.
    Note 3: The due date for this problem has been extended to October 13, 2004

11. Section 6.5: Do the problems in the handout that I provided in class.
    Due: October 18, 2004

12. Section 7.1: 1, 2, 3a
    Due October 20, 2004
    Notes: For 1b, do n=2 only, and draw pictures. For 3a, see Chapter 1, Section 7, if necessary.
    

13. Section 7.2: 4b, 5, 6a
    Due October 22, 2004
    Notes: In 4b, “vanish” means “vanish a.e.” In 5a, O is open. In 5b, ~E^o means (~E)^o. You may use 6a to do problem 5.

14. Section 7.3: 10
    Due October 27, 2004
    Note: The due date for this assignment has been extended to October 27, 2004.

15. Section 7.4: 16, 18
    Due October 27, 2004

16. Section 7.5: 21
    Due October 29, 2004

17. Section 7.6: 25
    Due November 1, 2004
    Hint: You may assume that you have done problem 14.

18. Section 7.8: 31, 32, 33
    Due November 5, 2004
    Note: For problem 31, “no open set” means “no nonempty open set.”
    Note: For problem 33(a), replace “Lebesgue measure 1-1/n” with “Lebesgue measure at least 1-1/n.”
    Note: We will skip Section 7.7.

19. Section 7.8: 34
    Due November 8, 2004
    Hint: By problem 5, page 143, the interior of a set is open.

20. Section 10.1: 1, 5, 9
    Due November 10, 2004
    Note for 5(c): Assume A is nonempty.

21. Section 10.2: 13, 14
    Due November 12, 2004

22. Section 10.3: 17
    Due November 15, 2004

23. Section 10.3: 20
    Due November 24, 2004
    Note: The due date for this assignment has been extended. The due date was originally November 22, 2004.

24. Section 10.3: 18
    Due November 29, 2004
    Notes: Assume that the closure of T is a proper subset of X. Assume that f is always a bounded linear functional. Use Proposition 7 to do this problem.

25. Section 10.3: 19
    Due December 1, 2004
    Note: Assume that delta is greater than or equal to zero in Proposition 7.

26. Section 10.3: Do the problems in Handout IX.
    Due December 3, 2004

    Most recent assignments:

27. Section 10.3: 23a, 23b.
    Also do the following exercise: Let phi be the map that we discussed in class. Thus X is a normed vector space, and phi maps X to X^**. For all x in X and all f in X^*, phi of x maps f to f(x). Prove each of the following:
    1. phi is linear.
    2. phi is injective (i.e. one-to-one).
    3. phi[X] is a vector space.
    Due December 6, 2004.

28. Prove propositions 2, 3, and 8 from Handout XII.
    Due December 6, 2004

29. Do the problems in Handout XIII.
    Due 11:59 P.M., December 9, 2004

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Contact Person: Larry Peterson
E-mail: lawrence.peterson@und.nodak.edu
Phone: (701) 777-4609
Date of most recent update: 6 December 2004
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