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: 20101101, Steve Shkoller
Units/Lecture:
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, 48 ($43)
Search by ISBN on Amazon: 0821827839
Search by ISBN on Amazon: 0821827839
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 

Each topic requires approximately 2 weeks to cover  

L^{p} spaces: Basic inequalities of Jensen, Holder, Minkowski; Completeness; Continuous linear functionals and weak convergence; Approximation and the theory of mollification; Dual space of L^{p}; Integral operators and Young's inequality  
The Sobolev spaces H^{k}(Ω) k a nonnegative integer: Weak derivatives; Completeness of H^{k}(Ω); Approximation by smooth functions; Sobolev embedding theorem; Morrey's inequality and the GagliardoNirenbergSobolev inequality; Extension and trace theorems; Rellich's theorem and weak compactness  
The Fourier transform: L^{1}(R^{n}) Fourier transform and its inversion; Schwartz functions of rapid decay; The Gaussian; Extension of Fourier transform to L^{2}(R^{n}) 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 H^{s}, s real: H^{s}(R^{n}) via the Fourier transform: Negativeorder spaces as distributions. H^{s}(T^{n}): Fourier series revisited 