Mathematics 421/510: Functional Analysis
Spring 2005
Instructor: I. Laba.
Office: Math Bldg 239.
Phone: 822 2450.
E-mail: ilaba(at)math.ubc.ca.
Office hours: Mon 3-4, Wed 11-12, Fri 1-2.
Homeworks and other handouts will be available
here.
Tentative outline:
- Point set topology background: topological spaces, continuous functions,
weak topology, compactness, local compactness (1-2 weeks)
- Banach spaces: linear functionals, dual spaces, Hahn-Banach
theorem, uniform boundedness, open mapping and closed graph theorems,
linear operators, operator spaces, strong, weak and weak* topology (3-4 weeks)
- Applications to real analysis: Lp spaces as
Banach spaces, duality, interpolation (1-2 weeks)
- Hilbert spaces: bases, Parseval's identity, self-duality,
projections, bounded and compact operators, self-adjoint operators, eigenvalues
and eigenvectors, spectral theorem for compact operators (3-4 weeks)
- Applications to Fourier analysis: L2 space as
a Hilbert space, Fourier series of periodic functions, Schwartz functions,
convolution, Fourier transform, Hausdorff-Young inequality, inversion formula,
summability issues if time allows (2-3 weeks)
Prerequisite:
Completion of Math 420/507, or equivalent familiarity with
basic real analysis, measure theory and integration (see the Math 420/507
syllabus for a list of topics).
Grading:
There will be several (5-6) problem sets throughout the semester. There
will be no final exam.
Sources:
- The main textbook is Real Analysis: Modern techniques
and their applications by G. Folland, 2nd ed., Wiley & Sons, Inc., 1999.
(See also list of errata.)
This is the same textbook that is used for Math 420/507. It does not
include all of the topics listed above, so at times you will have to
either rely on your class notes or consult other books.
- Functional Analysis: An Introduction, by Y. Eidelman, V. Milman,
and A. Tsolomitis, will be published by the AMS in the Graduate Studies in
Mathematics series this December
(check the AMS bookstore website
for more details). I will likely use it as a supplementary resource.
- Recommended supplementary books for specific portions of the material
will be listed here as needed.