MATH 503: Functional Analysis

Course instructor: Xiantao Li

Office: 219C McAllister Building

Phone: 3-9081

Email: xli@math.psu.edu

Office hours:

•Mondays, 11:00 AM

•Wednesdays, 1:30 PM

•or by appointment.

Course description

This is an introductory graduate course in functional analysis. We will cover basic concepts and theorems in Banach space and Hilbert space. As applications we will introduce Sobolev spaces and solutions of some PDEs.

Main topics:

      Theory:

•Metric Spaces. Normed spaces. Banach spaces. Linear operators. Examples.

•Spaces of bounded linear operators. The uniform boundedness principle, and the open mapping theorem. 

•Bounded linear functionals. Dual spaces. The Hahn-Banach extension theorem.          Separation of convex sets.

•Spaces of continuous functions. Ascoli’s theorem, Stone-Weierstrass’ theorem.

•Hilbert spaces. Perpendicular projections. Orthonormal bases. Self-adjoint operators.

   Applications:

•Compact operators on a Hilbert space. Fredholm’s alternative. Spectrum and eigenfunctions
of a compact, self-adjoint operator. Applications to Sturm-Liouville boundary value problems.

•Weak derivatives. Sobolev spaces. Approximation by smooth functions.

•Applications to linear elliptic equations.

•Semigroups of linear operators; generators, resolvents. Application to evolution PDEs.

References

1.Introductory functional analysis with applications, Erwin Kreyszig. (One copy reserved at the Math Physics Library, call number QA320.K74).

2.Lecture Notes on Funcational Analysis and Linear PDEs, A. Bressan, http://www.math.psu.edu/bressan/PSPDF/fabook.pdf

Homework: Homework will be assigned and collected about every week.

Midterm:  a 90-minute midterm exam.

Final exam: a 2-hour final exam.

Final grade: The final grade is based on the Homework (40%), the Midterm (20%), and the Final Exam 40%.

Grading scale:

    90% - 100%  A

   80% - 89%    B

   70% - 79%    C

   60% - 69%    D

   59% or below F