Course Description
This is the second part of a graduate course on an introduction to
functional analysis.

Topics
 Review of fundamental theorems for Banach spaces. Selected applications:
Numerical integration, convergence of Fourier series,
BrambleHilbert Lemma,
LaxMilgram Theorem, Negative norms, fixedpoint theorems.
 Locally convex topological spaces. Weak topologies and weak convergence. Selected applications: DunfordPettis Theorem on L1 weak compactness,
weak lower semicontinuity, Young measures.
 Distributions with applications to linear partial differential equations.
 Gaussian measures on Banach spaces: basic concepts, stochastic integrals, and nonlinear transformations.

Textbook and References
For the standard topics, we will follow closely the textbook

J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990.
We will also occasionally consult the following references:
 J. BarrosNeto, An Introduction to the Theory of Distributions,
Reprint ed., Krieger, 1981.
 V. I. Bogachev, Gaussian Measures, Amer. Math. Soc., 1998.
 N. Dunford and J. T. Schwartz,
Linear Operators. Part I. General Theory, WileyInterscience, 1958.
 R. E. Edwards, Functional Analysis, Dover, 1994.
 W. Rudin, Functional Analysis, 2nd ed., McGrawHill, 1991.
 H. H. Schaefer and M. P. Wolff, Topological Vector Spaces,, 2nd ed., Springer, 1999.
 K. Yosida, Functional Analysis,, 6th ed., Springer, 2003.
Lecture notes on special topics and applications will be distributed in the class.

Homework There will be a few homework assignments.

Office Hours
By appointment.
Lecture Notes
Last updated by Bo Li on April 4, 2012.
