640:502 Real Analysis II (Spring, 2004)
Announcements:
- Final Exam: Monday, May 10, 4:00-7:00 (Hill 425)
Lecturer: Prof.
Roe Goodman
Text: (required) Gerald B. Foland, Real Analysis: Modern
Techniques and Their Applications (2nd ed.),
ISBN #0-471-31716-0,
Wiley-Interscience/John Wiley Sons, Inc., 1999.
This course is a continuation of 640:501 from Fall 2003. The
goal is to give an introduction to core topics in real and
functional analysis that every professional mathematician should know,
and to cover some of the written qualifying exam topics that were not
in 640:501 last semester.
Homework Assignments: The list of
homework problems is updated frequently (be
sure to clear any old cached copies from your browser before loading).
Topics: The course will cover material from Chapters 4-8 of
Folland's book:
- Topological Spaces
Basic properties, compact spaces, Stone-Weierstrass theorem
- Introduction to Functional Analysis
Normed vector spaces, Hahn-Banach theorem, bounded linear
transformations, Closed graph and Open mapping theorem, applications
of Baire category theorem, Hilbert spaces, topological vector spaces,
weak and weak* convergence
- Lp Spaces
Integral inequalities, duality, bounded integral operators
- Introduction to Fourier analysis
Schwartz space, convolutions, Fourier transform and Fourier series, Plancherel
theorem, Poisson summation formula, Lp and pointwise
convergence of Fourier series
- Integration on Locally Compact Spaces
Continuous functions and Radon measures on locally compact spaces,
dual of C(X), vague convergence of measures
Here is a lecture-by-lecture
syllabus.
Grading: There are weekly graded homework assignments, an
in-class midterm exam (closed book) on March 10, and a final exam
(closed book) on May 10.
Roe Goodman /
goodman@math.rutgers.edu / Revised April 23, 2004