Math 240 Web Site


240A Hmwk
240B Hmwk
240C Hmwk
Lecture Notes

Math 240C (Driver, Spring 2002) Real Analysis


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.

Grading: The course grade will be computed using 

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

Course Summary


Math 240A

bulletQuick review of limit operations including sums on arbitrary sets
bulletBasics of Metric spaces, Normed spaces (including dual spaces), Hilbert spaces and Topological spaces
bulletIntroduction to sigma - algebras, measurable functions, and measures
bulletConstruction of the integral from a measure on a sigma - algebra
bulletGeneral properties of the integral (Fatou's lemma, monotone convergence, Lebesgue dominated convergence, Tchebychev inequality, Jensen's inequality)
bulletProduct measures and the Fubini-Tonelli theorems
bulletLp spaces, Holder inequality, the dual of dual of Lp spaces.

Math 240B

bulletLocally compact Hausdorff spaces, Uryshon's Lemma, Tietze Extension Theorem, Partitions of Unity, Alexandrov's compactification, Uryshon's metrization theorem.
bulletDensity and approximation theorems including the use of convolution and the Stone Weierstrass theorem.
bullet Hilbert space theory including projection theorems and orthonormal bases.
bulletFourier series and integrals, Plancherel theorem.
bulletExistence of Lebesgue measure and other measures using the Daniell integral.
bulletRiesz Representation theorem for measures.
bulletRadon-Nikodym Theorem for Positive measures.

Math 240C

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

Possible further topics

bulletA little complex analysis
bulletDistribution theory and elliptic regularity
bulletSobolev spaces
bulletUnbounded operators and the Spectral Theorem for self-adjoint operators
bulletProperties of ordinary differential equations
bulletImplicit and Inverse function theorems
bulletDifferentiable manifolds


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