Fall 2019 MAT 551: Analysis III, Functional analysis
ScheduleMW 2:30-3:50 Physics P123
InstructorRobert Hough
Office hoursW 6-7 in MLC, F 9am-11pm in Math Tower 4-118.
Description Selections from the following topics. Partial differential equations in higher dimensions; Sobolev spaces, calculus of variations, characteristics, Cauchy problem, energy estimates, maximum principles, Harmonic analysis; singular integrals, Hausdorff measure, harmonic measure, Hardy spaces, Functional analysis; spectral theory, distributions, Banach algebras.
TextbookPeter Lax. Functional Analysis. John Wiley and Sons (2002).
Supplementary TextbookMichael Reed and Barry Simon. Functional Analysis. John Wiley and Sons (1980).
Homework Weekly problem sets will be assigned, and collected in class on Monday.
GradingThe course grade is based upon the written homework.

Syllabus/schedule (subject to change)
Mon 8/261. Linear spaces Lax Chapters 1-4
Wed 8/282. Hahn BanachHW due 9/2: p. 12 #4,5, p. 32 #1,2
Mon 9/23. Normed linear spaces Lax Chapter 5
Wed 9/44. Normed linear spaces HW due 9/9: p.39 #3, p. 44 #4, p. 48 #8
Mon 9/95. Hilbert Space Lax Chapters 6-7
Wed 9/116. Applications of Hilbert Space HW due 9/16: p.53 #1, p.54 #3, p.62 #10, p.65 #2
Mon 9/167. Duality Lax Chapters 8-9
Wed 9/188. Applications of duality HW due 9/23: p.76 #1,2, p.77 #3,4, p.84 #6
Mon 9/239. Weak convergenceLax Chapters 10-12
Wed 9/2510. Weak and weak* topologies HW due 9/30: p.101 #2, p.104 #4, p.106 #6
Mon 9/3011. Convexity Lax Chapters 13-14
Wed 10/212. Extreme points HW due 10/7: p.122 #1,2, p.129 #7, p.144 #4, p.150 #5, p.152 #6
Mon 10/713. Bounded linear maps Lax Chapters 15-16
Wed 10/914. Examples of bounded linear maps HW due 10/16: p.164 #2, p. 165 #5, p.166 #6, p.171 #12, p.184 #3, p.185 #4
Mon 10/14 No class - Fall Break
Wed 10/1615. Banach algebrasLax Chapter 17, HW due 10/21: p.201 #1-5
Mon 10/2116. Spectrum Lax Chapters 19-20
Wed 10/2317. Examples of operators HW due 10/28: p.214 #2, p.219 #3, p.229 #1, p.230 #2,3, p.231 #4,6, p.232 #7
Mon 10/2818. Compact operators Lax Chapters 21-23
Wed 10/3019. Examples of compact operators HW due 11/4: p.234 #2, p.240 #4, p.241 #5, p.242 #6, p.243 #8
Mon 11/420. Fredholm Theory Lax Chapters 24-26
Wed 11/621. Invariant subspaces HW due 11/11: p.265 #2, p.271 #6
Mon 11/1122. Index theory, compact symmetric operatorsLax Chapters 27-29
Wed 11/1323. Examples HW due 11/18: p.312 #1, p.313 #2,3, p.318 #4, 5, p.321 #7, 8, p.327 #2
Mon 11/1824. Trace class, trace formulaLax Chapter 30
Wed 11/2025. Examples HW due 11/25: p.330 #2, p.333 #4, p.342 #5, p.347 #8
Mon 11/2526. Spectral theoryLax Chapter 31
Wed 11/27 No class - Thanksgiving
Mon 12/227. Self-adjoint operatorsLax Chapters 32-33
Wed 12/428. Examples HW due 12/9: p.372 10-12, p.390 #1, p.405 #6,7

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