18.155 | Fall 2004 | Graduate

Differential Analysis


Course Meeting Times

Lectures: 2 sessions / week, 1 hour / session


Fourier Analysis - Theory and Applications (18.103)


Rudin, W. Real and Complex Analysis. 3rd ed. New York, NY: McGraw-Hill, 1987. ISBN: 0070542341.


The course materials were presented according to the following breakdown of lectures:

  • Five lectures on measure and integration - Leading to Riesz representation theorem
  • Seven lectures on distributions and Fourier transform - Including Schwart space, Sobolev spaces and Sobolev embedding
  • Seven lectures on differential operators with constant coefficients, fundamental solutions and hypoellipticity
  • Four lectures on operators, trace class, Hilbert-Schmid


To pass the course each student was required to carry out one of the projects which were described in the second week of classes. The following projects were suggested to the students:

  1. Radon-Nikodym theorem
  2. Kuiper’s theorem: The group of unitary operators on a (separable infinite dimensional) Hilbert space is contractible
  3. Seeley’s extension theorem
  4. Gibb’s phenomenon
  5. Surjectivity of any non-trivial constant coefficient differential operator, P: S’ (Rn) → S’ (Rn)
  6. Every elliptic differential operator with constant coefficients is surjective as a map on C(U), for any open set U ⊂ Rn
  7. Lidskii’s theorem on trace class operators on L2(Rn)


The final grade was based on the homework and the project. There were no tests or examinations.

Homework 50%
Project 50%

Course Info

As Taught In
Fall 2004
Learning Resource Types
Lecture Notes
Problem Sets with Solutions