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, Steve 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:

201A, 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
Each topic requires approximately 2 weeks to cover Chapter 1 of Lieb and Loss, and Chapters 7 - 9, 13 of Applied Analysis


Basic measure and integration theory: Fundamental definitions from measure theory (proofs left to 206); Measurable functions and approximation by simple functions; Dominated and monotone convergence theorems and Fatou's Lemma; Fubini and Tonelli theorems; Definition of Lp and lp spaces and concrete examples of L2 Hilbert spaces


Fourier Series: Definitions and properties; Sobolev spaces Hs of periodic functions on torus for s real; Poisson summation/integral formula for the disk and the Dirichlet problem


Bounded linear operators on Hilbert space: Orthogonal projections; Dual space of Hilbert space and representation theorems; Weak convergence in Hilbert space, Banach-Alaoglu Theorem


Spectrum of bounded linear operators: Diagonalization of matrices; Spectral theorem for compact, self-adjoint operators; Compact operators; Fredholm Alternative Theorem; Functions of operators


Calculus on Banach Space: Bochner integrals; Derivatives of maps on Banach spaces; The calculus of variations