Office Hours: TBD.
LECTURE: MWF 0900 -- 0950 BAT 250 CRN 30243 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.
Textbooks: Carothers, Real Analysis and Kolmogorov & Fomin, Introduction to Real Analysis.
Reference for General Measure Spaces and Product Measure: Royden, Real Analysis.
Grading: Homework 50%, MidTerm Exam 20%, Final Exam 30%.
Midterm To be scheduled.
Final Exam March 19, 9:30-11:30am.
An optional discussion session is scheduled for Wednesdays at 1200 in WNGR 149A.
SCHEDULE
1. Lebesgue Measure in Rn C: Chap. 16 pp. 263-268. (01/05)
2. outer measure C: pp.268-274; K&F: 25.1.
3. measurable sets C: pp.292-293, 277-282; K&F: 25.2.
4. Lebesgue measure C: pp.283-286. (01/12)
Exercise 1: Show the conditions in Theorem 16.21 are equivalent and also characterize Caratheodory's definition of measurability.
5. General Measure Space and C: pp. 289-291.
6. Measurable functions C: Chap. 17 pp. 296-299.
7. MLK: no class meeting. (01/19)
8. C: pp. 300-304; K&F: 28.
9. C: pp. 306-307,
Exercise 2: Chapter 17, problems 5, 12, 25, 31, 42. Due 1/28.
10. C: pp. 305, 308-311. (01/26)
11. Integration C: pp. 312-314.
12. Monotone convergence theorem, Fatou's lemma C: 315-321.
13. C: pp. 315, 319, 322-323. (02/02)
Exercise 3: Chapter 18: problems 2, 4, 6, 10, 16. Due 2/11.
14. L^1 C: pp. 323-328.
15. Dominated convergence theorem C: pp. 328-331.
16. Lp C: pp. 342-344. (02/09)
17. pp. 344-349.
18. Complete; continuous operators.
19. (02/16)
20. Mid Term
21. Riemann integral.
22. Convergence in measure.C: pp. 337-342. (02/23)
23. General Measure Construction K&F: 26, 27.
24. Extension of measure.
25. Product Measure K&F: 35. (03/02)
26. measure by Sections. Exercise 4 due Wednesday March 11.
27. Fubini & Tonelli.
28. Review
29. Approximation: pp. 333-334,350-351.
30. Previews.