Analysis II: Measure, Integration and Banach Spaces

Math 114 / 10-11:30 Tu Th / Science Center 507
Harvard University -- Fall 2014

Instructor: Curtis T McMullen

  • Required: Royden and Fitzpatrick, Real Analysis, 4rd ed. Pearson, 2010.
  • Recommended: Stein and Shakarchi, Fourier Analysis. Princeton University Press, 2003.
  • Also useful: Oxtoby, Measure and Category. Springer-Verlag, 1980.
Prerequisites: Mathematics 23, 25, 55 or 112. Math 113 and 131 (Complex Analysis and Topology) recommended.

Topics. This course will provide a rigorous introduction to measurable functions, Lebesgue integration, Banach spaces and duality. Possible topics include:
  • Measure and Integration
    • Real numbers; open sets; Borel sets.
    • Measurable functions. Littlewood's 3 principles.
    • Lebesgue integration
    • Monotonicity, bounded variation, absolute continuity.
    • Differentiable and convex functions.
  • Banach spaces
    • The classical Banach spaces.
    • Hilbert spaces and Fourier analysis.
    • Topological spaces; C(X).
    • Compactness; Arzela-Ascoli.
  • Functional Analysis
    • Hahn-Banach theorem.
    • Baire category.
    • Open mapping theorem, closed graph theorem, uniform boundedness principle.
    • Weak topologies, Alaoglu's theorem.
    • Measures as the dual of C(X).
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. Grades will be based on homework (40%), an in-class midterm (20%) and a take-home final (40%).

Homework. Homework will be assigned every week. Late homework will not be accepted, however your lowest homework score will be dropped. Collaboration between students is encouraged, but you must write your own solutions, understand them and give credit to your collaborators. Please use only the course texts, course web page and your own lecture notes for homework.

Midterm. There will be an in-class midterm, covering the first half of the course. It will be given in approximately the 7th week of the course.

Final. There will be a take-home final exam, to be completed during reading period.

Calendar 2014.
  • Tu, 2 Sep. First class
  • Th-F, 27-28 Nov. Thanksgiving
  • Tu, 2 Dec. Last class
  • Th-W, 4-10 Dec. Reading period

Course home page: