Office Hours: TBD.
LECTURE: MWF 0900 -- 0950 BAT 250 CRN 50527 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.
(413/513) Bounded variation, absolute continuity, differentiation, Radon-Nikodym theorem, Riemann-Lebesgue-Stieltjes integrals, Banach and Hilbert spaces, characterization of dual spaces, applications.
Textbooks: Carothers, Real Analysis and Kolmogorov & Fomin, Introduction to Real Analysis.
Reference: Royden, Real Analysis.
Hilbert Space Methods
Study session: Mondays at noon, BEXL 102.
Grading: Homework 50%, MidTerm Exam 20%, Final Exam 30%.
Midterm To be scheduled.
Final Exam Monday, June 8, 1800.
SCHEDULE
1. No Class meeting. (03/30)
2. No Class meeting.
3. Bounded variation C: pp.202-206.
4. monotone functions K&F: sec. 31.1; C: pp.207-208; 32-33. p. 33: Exc. 36. (04/06)
5. Vitali covering theorem C: 287-288 (S. Banach's proof).
6. Lebesgue differentiation theorem C: 359-365; K&F: sec. 31.2 (different!).
7. absolute continuity C: 366-371. (04/13)
8. antiderivative C: 366-371; K&F: sec. 33.
9. FTC C: 372-376.
10. linear algebra HSM (Hilbert Space Methods): pp. 1-5; K&F: sec. 13. (04/20)
11. seminormed spaces HSM: pp. 6-8.
12. linear operators HSM: pp. 8 - 11.
Exercises 1: C: 2.36, 20.11; HSM: 1.7 with f in L^1(G). Show f -> T^f is 1-1. Due 5/1.
13. completion HSM: pp. 11-13. (4/27)
14. Uniform-boundedness, Open-mapping theorems
15. Closed-graph, Closed-range theorems. Partially ordered sets K&F: sec. 3.1, 3.7.
16. Hahn-Banach K&F: sec. 14.4 (p.132),18.3 (p.180). (5/04)
17. Dual spaces K&F: sec. 19.4 (p.190-193) Beware of missprints!
18. Hilbert space, Minimization K&F: sec. 16.1-2, 16.8; HSM: pp. 13-15.
Exercises 2: HSM: 2.6 (p.27) [NOTE: W = R in second part.], 4.3 (p. 28), Thm 3.Ca,b (p. 12); K&F: pp 334-5 missing details. Due (5/15).
19. directional derivative, H'= H, projection (5/11)
20. dual of L^p(0,1)
21. absolutely continuous measure
22. Radon-Nikodym Theorem (5/18)
23. signed measures K&F: sec 34
24. Jordan decomposition
Memorial Day (5/25)
26. dual of L^p(S)
Exercise 3: Show directly that lp'=(lp)'. Due (6/1).
27. Fourier series HSM: Sec. 7.1, pp. 24-25. (Read K&F: sec. 16.)
28. expansion theory for compact self-adjoint operators HSM: Sec. 7.2. (6/1)
29. generalized derivatives
30. BVPs & orthonormal bases of L^2 Read C: chap 15 and 352-356.