MATH 241B: Functional Analysis (II)
Spring quarter, 2012

Instructor: Bo Li
(Office: AP&M 5723. Phone: (858) 534-6932. Email:

Class time and place: 4:00 - 5:15 Tuesdays and Thursdays, AP&M 5829.

Course Description

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


  • Review of fundamental theorems for Banach spaces. Selected applications: Numerical integration, convergence of Fourier series, Bramble-Hilbert Lemma, Lax-Milgram Theorem, Negative norms, fixed-point theorems.
  • Locally convex topological spaces. Weak topologies and weak convergence. Selected applications: Dunford-Pettis 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. Barros-Neto, 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, Wiley-Interscience, 1958.
  • R. E. Edwards, Functional Analysis, Dover, 1994.
  • W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, 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.


There will be a few homework assignments.

Office Hours

By appointment.

Lecture Notes

Last updated by Bo Li on April 4, 2012.