Department Syllabus
MAT 201B: Analysis

When taught: Winter, every year
Suggested texts ($): Analysis by Elliott H. Lieb and Michael Loss, Chapter 1, ISBN-10: 0-8218-2783-9 ($43), and Applied Analysis by Hunter and Nachtergaele, Chapters 7 - 9, 13. Available here: (http://www.math.ucdavis.edu/~hunter/book/pdfbook.html)
Units/lectures: 4 units; lecture/discussion section
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

Prepared by: Steve Shkoller
Approved by GPC: November 2010

LecturesSections Topics/Comments
Each topic requires approximately 2 weeks to coverChapter 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