Real Analysis
Math 212a -- Tu Th 10-11:30 -- 112 SC
Harvard University -- Fall 2003
Instructor:
Curtis T McMullen
Course Assistant:
Jonathan Kaplan
(jkaplan@math.harvard.edu)
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.
Calendar.
- 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