MATH 245C : Real Analysis

  1. Course description: Continuation of 245B.  Interpolation theory, Fourier analysis, Sobolev spaces, distribution theory.  Hausdorff dimension may be covered if time permits.


  1. This class will be a direct continuation of 245B.  A passing grade in 245B (or equivalent) will be a mandatory prerequisite for enrollment in this class.
  2. The first class is on Monday, March 28.  We will begin with 245B Notes 12: Continuous functions on locally compact Hausdorff spaces, and then go through the 245C notes as time permits.
  3. (Mar 30)  Beginning with the next class (i.e. Friday April 1), the class will be held at MS 5128 rather than MS 6229.
  4. (Apr 19) A second office hour has been set aside at Tu 11-12 (since some students were not able to make the W 1-2 office hour).
  5. (Apr 22) Due to the second distinguished lecture of Ehud Hrushovski, there will be NO CLASS on Wednesday, Apr 27.

  1. Instructor: Terence Tao,, x64844, MS 6183
  2. Lecture: MWF 2-2:50, MS 5128 (note change of room)
  3. Quiz section: N/A
  4. Office Hours: Tu 11-12; W 1-2
  5. TA: N/A
  6. TA Office hours: N/A
  7. Textbook: Folland's "Real analysis", and my “An epsilon of room, Vol. I.”, available at  
  8. Prerequisites: A passing grade in 245B (or equivalent) is a mandatory prerequisite for this course.  (This course will be a direct continuation of the winter quarter 245B course.)
  9. Grading: Grading will be based on homework and attendance.
  10. Reading Assignment: We will cover most of Chapters 1.10-1.15 of “An epsilon of room, Vol. I”.
  11. Homework: Approximately four homework assignments will be given, mostly from “An epsilon of room, Vol. I”.
  1. First homework assignment (due Monday, Apr 11):
  1. Exercise 1.10.9 (Tietze extension theorem for unbounded functions).
  2. Exercise 1.10.17 (M(X) as dual of C_c(X) and C_0(X)).  Note that in this section, X is always assumed to be LCH and sigma-compact.
  3. Exercise 1.10.25 (Density of Fourier series).  Note an important typo in this question: [R,Z] (or [0,1]) should be R/Z (i.e. the unit circle).
  1. Second homework assignment (due Monday, Apr 25)
  1. Exercise 1.11.4 (Phragmen-Lindehof principle)
  2. Exercise 1.11.8 (Weak and strong L^p equivalent up to logarithmic factors).  Important typo: log(1+|X|) should be log1/p(1+|X|).
  3. Exercise 1.11.14 (L^p characterisation of exponential integrability)
  1. Third homework assignment (due Monday, May 9):
  1. Exercise 1.11.11 (Characterisation of weak L^p)
  2. Exercise 1.12.6 (Existence of Haar measure, compact case). You may use Tychonoff’s theorem (Theorem 1.8.14) without proof.
  3. Exercise 1.12.16 (Pontryagin dual of compact group is discrete)
  1. Fourth homework assignment (due Monday, May 23):
  1. Exercise 1.12.20 (Convergence of Fourier series)
  2. Exercise 1.12.41 (Poisson summation formula)
  3. Exercise 1.13.16 (Division by x)
  1. Exams: There is no examination for this course.

Online resources:

  1. The blog for this course.