Syllabus Detail

Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 201B: Analysis

Approved: 2010-11-01 (revised 2025-05-30, Hunter/Fraas/Shkoller)
Units/Lecture:
Winter, every year; 4 units; lecture/discussion section
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Analysis by Elliott H. Lieb and Michael Loss, Chapter 1 ($43), and Applied Analysis by Hunter and Nachtergaele, Chapters 7 - 9, 13, available at provided link.
http://www.math.ucdavis.edu/~hunter/book/pdfbook.html
Search by ISBN on Amazon: 0-8218-2783-9
Prerequisites:
Graduate standing in Mathematics or Applied Mathematics, or consent of instructor.
Course Description:
Metric and normed spaces. Continuous functions. Topological, Hilbert, and Banach spaces. Fourier series. Spectrum of bounded and compact linear operators. Linear differential operators and Green's functions. Distributions. Fourier transform. Measure theory. Lp and Sobolev spaces. Differential calculus and variational methods.
Suggested Schedule:
Lectures Sections Topics/Comments
7   Fourier series: Motivation and definition of Fourier series, illustrative examples, convergence in L2, pointwise and uniform convergence, convolution, decay of Fourier coefficients, and applications (heat and/or wave equation).
5   Lp spaces: Monotone Convergence Theorem, Dominated Convergence Theorem, examples illustrating convergence of sequences and
integrals, definition and completeness of Lp spaces, approximation by simple and smooth functions, and Fubini’s theorem.
7-8   Fourier transform (FT): Definition of the Fourier transform, Fourier transforms of Gaussian and other examples, Schwartz space, inverse Fourier transform, translation and differentiation properties, Fourier transform in L2, Plancherel theorem, tempered distributions, convolutions, and applications (e.g., Poisson equation and Green’s functions).
6-7   Spectral theory: Definitions of spectrum and resolvent sets, basic spectral properties, illustrative examples, spectrum of self adjoint operators, classification of spectra, multiplication operators, unitary operators, spectral theorem for self-adjoint operators (proof omitted), spectral theory of the discrete Laplacian using Fourier series, and spectral theory of compact operators (optional).