240A Hmwk
240B Hmwk
240C Hmwk
Lecture Notes
Announcements

## 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," 2nd 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.

## 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 Lp spaces, Holder inequality, the dual of dual of Lp 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

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