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 201A: Analysis
Approved: 2010-11-01, Steve Shkoller

Units/Lecture:

Fall, every year; 4 units; lecture/discussion section

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Applied Analysis by Hunter and Nachtergaele, Chapters 1, 3-6
http://www.math.ucdavis.edu/~hunter/book/pdfbook.html

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 Chapters 1, 2, and 4-6.


Metric and normed spaces (review): Metrics, norms, limits, liminf, limsup; Pointwise, uniform, and norm convergence; Continuity and completeness; Compactness in finite-dimensions; Compact and locally compact spaces.


Spaces of continuous functions: Definition of spaces; Convergence in the uniform topology; Tychonoff's Theorem; Arzela-Ascoli Theorem; Stone-Weierstrass Theorem


Topological spaces: Definition of topological spaces; Bases of open sets; Comparing topologies


Banach spaces: Normed vector spaces; Linear functionals and bounded linear maps; The kernel and range of linear maps; Convergence in the space of bounded linear operators; Dual spaces


Hilbert spaces: Inner products; Orthogonality and projections; Orthonormal bases; Applications