Office Hours: M 1130 - 1230, F 1330 - 1430.
LECTURE: MWF 0900 -- 0950 BAT 250 CRN 16007 Instructor: R.E. Showalter Kidder 298b show@math.oregonstate.edu
Topics: (411/511) Topological concepts in metric, normed, and inner-product spaces. Properties of continuous functions, including the Stone-Weierstrass theorem. Introduction to function spaces, contraction mappings, fixed points, and applications.
(412/512) Lebesgue measure and integration in one and several variables, basic convergence theorems, Lebesgue spaces, Fubini's theorem, and applications to Fourier transforms and probability.
Textbook: Carothers, Real Analysis and Kolmogorov, Introduction to Real Analysis.
Grading: Homework 50%, MidTerm Exam 20%, Final Exam 30%.
An optional discussion session is scheduled for Wednesdays at 1600 in BAT 250.
SCHEDULE
1. Chap 1 - Review: Read pp. 3-14. (9/29/08)
2. Chap 3 - Metric Spaces: Read pp. 15, 36-39.
3. Normed linear space: pp. 40 - 44.
4. Limits: pp. 45-50. (10/6)
5. Chap 2 - Cardinality: pp. 18-22.
6. power set, Bernstein theorem, Cantor set.
HMWK due Monday (10/13): Chap 1: 3, 13, 26; Chap 3: 2, 40.
7. Chap 4 - Open sets (10/13)
8.
9.
10. Chap 5 - Continuity: pp. 63-68. (10/20)
11. Homeomorphism, C[a,b]: pp. 69-77.
12. Chap 6 - Connectedness.
HMWK due Wednesday (10/29): Chap 3: 36, 37, 39; Chap 4: 5, 17, 20, 21.
13. Chap 7 - Totally bounded sets. (10/27).
14. Completeness: pp. 92-97.
15. Completions.
16. Fixed points: pp. 98-102. (11/03).
17. MidTerm
18. Recap
19. Chap 8 - Compactness: pp. 108-113. (11/10).
20. Uniform continiuty: pp. 114-119.
21. Read pp. 120-126.
HMWK due Monday (11/17): p. 102:42, 109:9, 110:12, and for example of pp.101-102, show some power of F is a strict contraction on C[0,1].
22. Chap 9 - Category. (11/17)
23. Chap 10 - Sequences of functions: pp. 139-154.
24. Chap 11 - Equicontinuity; Ascoli-Arzela: pp. 178 - 182.
HMWK due Monday (11/24): chap 8: 23, 80, 81; chap 9: 2, 4, 22.
25. Equicontinuity and Boundedness. (11/24)
26. No Class
27. Chap 12 - Algebras and Lattices: pp. 188 - 193. (12/01)
28. Stone-Weierstrass Thm: pp. 194-197.
29. pp. 198-200.
Final Exam: Wednesday 12/10 1800 BAT 250.