Math 240C -- Real Analysis (Spring 2004) Course Information

(http://math.ucsd.edu/~driver/240A-C-03-04/index.htm)

New Announcements

 Final exam results and solutions for Winter 04 can be had by clicking on the following link: Test Data.

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

TA:                     Matt Cecil  (mcecil@math.ucsd.edu). Office: APM 6402A.

Meeting times: MWF 12:00 -- 12:50 PM in  APM 7421.

Textbook:         I will mainly follow the lecture notes which will be available from this web-site.  We will also use Gerald B. Folland, "Real Analysis, "Modern Techniques and Their Applications," 2nd edition.

Prerequisites: Students are assumed to have taken at the very least a two-quarter sequence in undergraduate real analysis covering in a rigorous manner the theory of limits, continuity and the like in Euclidean spaces and general metric spaces. The theorems of Heine-Borel (compactness in Euclidean spaces), Bolzano-Weierstrass (existence of convergent subsequences), the theory of uniform convergence, Riemann integration theory should have been covered. One quarter of undergraduate complex analysis is also recommended.

Homework:    There will be weekly Homework assignments.

Test times:     There will be one midterm a little before the qualifying exam for the course. This will also serve as another practice session for the qualifying exam.

Office Hours:  To be determined. (Feel free to stop in whenever you can find me.)

## Tentative Course Outline

Math 240A

 Review of limit operations including sums on arbitrary sets Basics of Metric spaces, Normed spaces (including dual spaces), Topological spaces Riemann Integral on Banach Spaces, Linear ODE, Classical Weierstrass approximation  theorem Introduction to sigma - algebras, measurable functions, and measures The structure of measurable sets and functions, e.g. Dynkin's Multiplicative System Theorem Construction of the integral from a measure General properties of the integral (Fatou's lemma, monotone convergence, Lebesgue dominated convergence, Tchebychev inequality, Jensen's inequality)

Math 240B

 Product measures and the Fubini-Tonelli theorems Lebesgue Measure on Rd  and the change of variables theorem Lp spaces, Holder inequality, the dual of dual of Lp spaces. Topology: connectedness, compactness, and countability axioms More Topology: 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.

Math 240C

 Existence of Lebesgue measure and other measures via the Riesz-Markov theorem Radon-Nikodym Theorem for Positive measures. 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 Riesz Representation theorem for measures. 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 A little distribution theory and elliptic regularity

Possible further topics

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

Math 240B -- Real Analysis (Winter 2004) Course Information

(http://math.ucsd.edu/~driver/240A-C-03-04/index.htm)

New Announcements

 Final exam results and solutions for Winter 04 can be had by clicking on the following link: Test Data.

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

TA:                     Matt Cecil  (mcecil@math.ucsd.edu). Office: APM 6402A.

Meeting times: MWF 12:00 -- 12:50 PM in  APM 7421.

Textbook:         I will mainly follow the lecture notes which will be available from this web-site.  We will also use Gerald B. Folland, "Real Analysis, "Modern Techniques and Their Applications," 2nd edition.

Prerequisites: Students are assumed to have taken at the very least a two-quarter sequence in undergraduate real analysis covering in a rigorous manner the theory of limits, continuity and the like in Euclidean spaces and general metric spaces. The theorems of Heine-Borel (compactness in Euclidean spaces), Bolzano-Weierstrass (existence of convergent subsequences), the theory of uniform convergence, Riemann integration theory should have been covered. One quarter of undergraduate complex analysis is also recommended.

Homework:    There will be weekly Homework assignments.

Test times:     There will be one midterm given sometime during the week 5 or 6 of the quarter.
The final is scheduled for Monday, December 8 from 11:30 - 2:30 PM.

Office Hours:  W. & F. at 1:30 - 2:30 PM in my office, and Mondays 5:30- 6:30 PM in APM 7421.