Math 240C (Driver, Spring 2002) Real Analysis

(http://math.ucsd.edu/courses.html)

Instructor: Bruce Driver, APM 7414, 534-2648.

Meeting times: MWF 11:15 -- 12:05 PM in APM 7421.

Textbook: Gerald B. Folland, "Real Analysis, "Modern Techniques and Their Applications," 2^nd edition. I will also make lecture notes available from this site.

Prerequisites: Students are assumed to have completed last quarters Math 240B.

Homework: There will be weekly homework assignments, which I will grade.

Test times: The final is scheduled for ????.

Office Hours: To be determined.

Grading: The course grade will be computed using

Grade=.3(Home Work)+.3(Midterm)+.4(Final).

Course Summary

Math 240A

	Quick review of limit operations including sums on arbitrary sets
	Basics of Metric spaces, Normed spaces (including dual spaces), Hilbert spaces and Topological spaces
	Introduction to sigma - algebras, measurable functions, and measures
	Construction of the integral from a measure on a sigma - algebra
	General properties of the integral (Fatou's lemma, monotone convergence, Lebesgue dominated convergence, Tchebychev inequality, Jensen's inequality)
	Product measures and the Fubini-Tonelli theorems
	L^p spaces, Holder inequality, the dual of dual of L^p spaces.

Math 240B

	Locally compact Hausdorff spaces, Uryshon's Lemma, Tietze Extension Theorem, Partitions of Unity, Alexandrov's compactification, Uryshon's metrization theorem.
	Density and approximation theorems including the use of convolution and the Stone Weierstrass theorem.
	Hilbert space theory including projection theorems and orthonormal bases.
	Fourier series and integrals, Plancherel theorem.
	Existence of Lebesgue measure and other measures using the Daniell integral.
	Riesz Representation theorem for measures.
	Radon-Nikodym Theorem for Positive measures.

Math 240C

	Signed and complex measures, Hahn decomposition, Jordan decomposition and the Radon - Nikodym theorem
	Differentiation of measures on R^n and the fundamental theorem of calculus
	More Banach space results: Banach Steinhaus Theorem, Hahn Banach Theorem, Open mapping theorem and the closed graph theorem
	Some Calculus on Banach spaces and the change of variable theorem
	Fourier Transform and its properties with basic applications to PDE and Sobolev spaces.
	The Spectral Theorem for bounded self-adjoint operators on a Hilbert space.

Possible further topics

	A little complex analysis
	Distribution theory and elliptic regularity
	Sobolev spaces
	Unbounded operators and the Spectral Theorem for self-adjoint operators
	Properties of ordinary differential equations
	Implicit and Inverse function theorems
	Differentiable manifolds

Jump to Bruce Driver's Homepage. Go to list of mathematics course pages.

Last modified on March 19, 2002 at 01:49 PM.