Real Analysis
Math 212a / MWF 121 / Science Center 309
Harvard University  Fall 1998
Instructor:
Curtis T McMullen
(ctm@math.harvard.edu)
Course Assistant:
Spiro Karigiannis
(spiro@math.harvard.edu)
Required Texts
 Royden,
Real Analysis, 3rd ed.
PrenticeHall, 1988
 Rudin,
Functional Analysis, 2nd ed.
McGrawHill, 1991
Recommended Texts
 Oxtoby,
Measure and Category.
SpringerVerlag, 1980.
 Berberian,
Lectures on Functional Analysis.
SpringerVerlag, 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; ArzelaAscoli.

HahnBanach 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 takehome 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 takehome 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.
Calendar.
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
1115 Jan (MF): Takehome final
16 Jan (Sat): Exams begin
