Real Analysis

Math 212a -- Tu Th 10-11:30 -- 112 SC
Harvard University -- Fall 2003

Instructor: Curtis T McMullen

Course Assistant: Jonathan Kaplan (

Required Texts
  • Royden, Real Analysis, 3rd ed. Prentice-Hall, 1988. Errata
  • Rudin, Functional Analysis, 2nd ed. McGraw-Hill, 1991
Recommended Texts
  • Oxtoby, Measure and Category. Springer-Verlag, 1980.
Prerequisites. Intended for graduate students. Undergraduates require Math 113, 131 and permission of the instructor.

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.
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.

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 an extended homework assignment at the end of the course in place of a final exam. Collaboration on the final is not permitted, but you may refer to your course notes and the texts for the course.

  • Tu, 16 Sep. First class
  • Tu, 11 Nov. Veterans day
  • Th-F, 27-28 Nov. Thanksgiving
  • Tu, 16 Dec. Last class
  • M-F, 5-16 Jan. Reading period

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