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 201C: Analysis

Approved: 2010-11-01, Steve Shkoller
Spring, every year; 4 units; lecture/discussion section
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Lecture notes to be supplied by Steve Shkoller or Analysis by Elliott H. Lieb and Michael Loss, Chapter 2, 4-8 ($43)
Search by ISBN on Amazon: 0-8218-2783-9
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
Each topic requires approximately 2 weeks to cover

Lp spaces: Basic inequalities of Jensen, Holder, Minkowski; Completeness; Continuous linear functionals and weak convergence; Approximation and the theory of mollification; Dual space of Lp; Integral operators and Young's inequality

The Sobolev spaces Hk(Ω) k a non-negative integer: Weak derivatives; Completeness of Hk(Ω); Approximation by smooth functions; Sobolev embedding theorem; Morrey's inequality and the Gagliardo-Nirenberg-Sobolev inequality; Extension and trace theorems; Rellich's theorem and weak compactness

The Fourier transform: L1(Rn) Fourier transform and its inversion; Schwartz functions of rapid decay; The Gaussian; Extension of Fourier transform to L2(Rn) and Plancheral's theorem; Tempered distributions S' and extension of Fourier transform to S'; Examples of the use of Fourier transform to Poisson, heat, and wave equations

The Sobolev spaces Hs, s real: Hs(Rn) via the Fourier transform: Negative-order spaces as distributions. Hs(Tn): Fourier series revisited