Spring 2020 MAT 533: Analysis II
ScheduleMW 10:00-11:20 am Math 4-130
InstructorRobert Hough
Office hoursF 9am-11pm in Math Tower 4-118, F 3-4pm in MLC.
Description Representations and decomposition theorems in measure theory; Fubini's theorem; L-p spaces; Fourier series; Laplace, heat and wave equations; open mapping and uniform boundedness theorems for Banach spaces; differentation of the integral; change of variable of integration.
TextbookGerald B. Folland. Real Analysis. John Wiley and Sons (1999).
Homework Weekly problem sets will be assigned, and collected in class on Monday.
GradingThe course grade is based upon the written homework (30%), a midterm (30%) and final exam (40%).

Syllabus/schedule (subject to change)
Mon 1/271.Review of point-set topology Folland 4.1-4.2, 4.4-4.5
Wed 1/292.Compact and locally compact spaces HW due 2/3: Homework 1, Solutions.
Mon 2/33.Ascoli-Arzela Theorem, Stone-Weierstrass Theorem Folland 4.6-4.7, 5.1-5.2
Wed 2/54. Normed vector spaces, linear functionals HW due 2/10: Homework 2, Solutions.
Mon 2/105. Baire Category Theorem Folland 5.3-5.4
Wed 2/126. Topological vector spaces HW due 2/17: Homework 3, Source file, Solutions.
Mon 2/177. Hilbert space Folland 5.5, 7.1
Wed 2/198. Positive linear functions HW due 2/24: Homework 4, Source file
Mon 2/249. Lusin's TheoremFolland 7.2-7.3
Wed 2/2610. The Riesz Representation Theorem HW due 3/2:
Mon 3/211. The Fubini-Tonelli Theorem Folland 7.4, 8.1
Wed 3/412. Fourier preliminaries HW due 3/9:
Mon 3/913. Convolutions Folland 8.2
Wed 3/11 Midterm I
Mon 3/16 No class - Spring Break
Wed 3/18 No class - Spring Break
Mon 3/2314. The Fourier transform Folland 8.3-8.4
Wed 3/2515. Fourier inversion on the torus HW due 3/30:
Mon 3/3016. Distributions Folland 8.5-8.6
Wed 4/117. Sobolev spaces HW due 4/6:
Mon 4/618. Applications to PDE Folland 8.7-8.8
Wed 4/819. Fourier analysis of measures HW due 4/13:
Mon 4/1320. Basic concepts of probability Folland 9.1-9.2
Wed 4/1521. The law of large numbers HW due 4/20:
Mon 4/2022. The central limit theorem Folland 9.3-9.4
Wed 4/2223. Construction of sample spaces HW due 4/22:
Mon 4/2724. Brownian motionFolland 9.5, 10.1
Wed 4/2925. Haar measure HW due 5/4:
Mon 5/426. Hausdorff measureFolland 10.2-10.3
Wed 5/627. Manifolds HW due 5/11:

Final Exam May 18, from 8am-10:45 am

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