Spring 2022 MAT 533: Analysis II
ScheduleMW 2:40-4:00 pm Physics P112
InstructorRobert Hough
Office hoursTh 9am-11pm in Math Tower 4-118, W 7-8pm 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. Submit the homework each week by Midnight on Monday to Matthew Dannenberg, matthew.dannenberg@stonybrook.edu.
GradingThe course grade is based upon the written homework (30%), a midterm (30%) and final exam (40%).

Syllabus/schedule (subject to change)
Mon 1/241.Review of point-set topology Folland 4.1-4.2 HW due 1/31: Homework 1.
Wed 1/262.Compact and locally compact spaces Folland 4.4-4.5
Mon 1/313.Ascoli-Arzela Theorem, Stone-Weierstrass Theorem Folland 4.6-4.7 HW due 2/7: Homework 2.
Wed 2/24. Normed vector spaces, linear functionals Folland 5.1-5.2
Mon 2/75. Baire Category Theorem Folland 5.3 HW due 2/14: Homework 3
Wed 2/96. Topological vector spaces Folland 5.4
Mon 2/147. Hilbert space Folland 5.5 HW due 2/21: Homework 4
Wed 2/168. Positive linear functions Folland 7.1
Mon 2/219. Lusin's TheoremFolland 7.2 HW due 2/28: Homework 5
Wed 2/2310. The Riesz Representation Theorem Folland 7.3
Mon 2/2811. The Fubini-Tonelli Theorem Folland 7.4 HW due 3/7: Homework 6
Wed 3/212. Fourier preliminariesFolland 8.1
Mon 3/713. Convolutions Folland 8.2 HW due 3/21: Homework 7
Wed 3/914. The Fourier Transform Folland 8.3
Mon 3/14 No class - Spring Break
Wed 3/16 No class - Spring Break
Mon 3/21 MidtermPractice Midterm
Wed 3/2315. Fourier inversion on the torus Folland 8.4
Mon 3/2816. Distributions Folland 8.5 HW due 4/4: Homework 8
Wed 3/3017. Sobolev spaces Folland 8.6
Mon 4/418. Applications to PDE Folland 8.7
Wed 4/619. Fourier analysis of measuresFolland 8.8
Mon 4/1120. Basic concepts of probability Folland 9.1 HW due 4/18: Homework 9
Wed 4/1321. The law of large numbersFolland 9.2
Mon 4/1822. The central limit theorem Folland 9.3 HW due 4/25: Homework 10
Wed 4/2023. Construction of sample spaces Folland 9.4
Mon 4/2524. Brownian motionFolland 9.5 HW due 5/2: Homework 11
Wed 4/2725. Haar measureFolland 10.1
Mon 5/226. Hausdorff MeasureFolland 10.2
Wed 5/427. Integration on manifoldsFolland 10.3

The Final Exam is May 18 11:15am-1:45pm. Practice Final Exam

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