Real Analysis

Math 212a / MWF 12-1 / Science Center 309
Harvard University - Fall 1998

Instructor: Curtis T McMullen (

Course Assistant: Spiro Karigiannis (

Required Texts
  • Royden, Real Analysis, 3rd ed. Prentice-Hall, 1988
  • Rudin, Functional Analysis, 2nd ed. McGraw-Hill, 1991
Recommended Texts
  • Oxtoby, Measure and Category. Springer-Verlag, 1980.
  • Berberian, Lectures on Functional Analysis. Springer-Verlag, 1974
Prerequisites. Intended for graduate students. Undergraduates require the permission of the instructor. Freshmen who wish to enroll must talk to the Director of Undergraduate Studies in Mathematics, Prof. Taubes.

Topics. This course will provide a rigorous introduction to measurable functions, Lebesgue integration, Banach spaces and duality. Possible topics include:
  • Functions of a real variable
    • Real numbers; open sets; Borel sets; transfinite induction.
    • Measurable functions. Littlewood's 3 principles.
    • Lebesgue integration
    • Monotonicity, bounded variation, absolute continuity.
    • Differentiable and convex functions.
    • The classical Banach spaces.
  • Banach spaces
    • Metric spaces.
    • Baire category.
    • Compactness; Arzela-Ascoli.
    • Hahn-Banach theorem.
    • Open mapping theorem, closed graph theorem.
    • Uniform boundedness principle.
    • Weak topologies, Alaoglu's theorem.
    • The space of measures.
    • Distributions.
    • Solutions of elliptic equations.
Reading and Lectures. Students are responsible for all topics covered in the readings and lectures. Assigned material should be read before coming to class. Lectures may go beyond the reading, and not every topic in the reading will be covered in class.

Grades. Graduate students who have passed their quals are excused from a grade for this course. Grades for other students will be based on homework (50%) and a take-home final (50%). Your lowest homework grade will be dropped.

Homework. Homework will be assigned once every week or two. Late homework will not be accepted. Collaboration between students is encouraged, but you must write your own solutions, understand them and give credit to your collaborators.

Final. There will be a take-home final, available at 12 pm on Monday, 11 January, and due by 4:00 pm on Friday, 15 January. The final should be picked up and returned to the Math Department Office, 325 Science Center. Collaboration on the final is not permitted, but you may refer to your course notes and the texts for the course.

16 Sept (W): First class
12 Oct (M): Columbus day -- no class
19 Oct (M): Last day to add or drop courses
11 Nov (W): Veteran's day -- no class
27 Nov (F): Thanksgiving -- no class
16 Dec (W): Last class
4 Jan (M): Reading period begins
11-15 Jan (M-F): Take-home final
16 Jan (Sat): Exams begin

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