This is the second half of the functional analysis course. The contents include: 1. Topological vector spaces, Locally convex spaces, Semi-normed linear space. 2. Introduction to generalized functions. 3. Unbounded operators. 4. Spectral theorem for unbounded operators 5. Semigroup theory and applications. 6. Infinite dimensional probability analysis (optional).

After completion of this course, students are expected to be familiar with some knowledge of abstract spaces, the interplay between the space and its dual. Unbounded operators are ubiquitous in many fields such as partial differential equations, harmonic analysis, etc. Techniques in infinite dimensional analysis are useful in statistics and machine learning.

Weekly homework and final exam

1. Methods of Modern Mathematical Physics I. Functional Analysis, by Michael Reed and Barry Simon, Academic Press, 1980. 2. Functional Analysis, Walter Rudin, New York : McGraw-Hill, [1990], c1991. 3. Functional analysis: An introduction (Pure and applied mathematics, v. 15), Ronald Larsen, M. Dekker, 1973.

1. Functional Analysis, by Peter Lax, John Wiley & Sons, Inc. 2002. 2. Functional Analysis, by Kosaku Yosida, Springer, 1980.