MATH 632. Functional Analysis (Fall 2006)
Meeting times: MWF, 9:00am-9:50am (MTH 0407)
Instructor: Professor Jonathan
Rosenberg. His office is room 2114 of the Math Building,
phone extension 55166, or you can contact him by
email. His office hours
are M and F 1-2, or by appointment.
Text:
Peter
Lax, Functional Analysis (Wiley). The book is available at a substantial
discount at
amazon.com and possibly other online bookstores,
so shop around.
Prerequisite: MATH 631 (real analysis).
Catalog description:
Introduction to functional analysis and operator theory: normed linear spaces,
basic principles of functional analysis, bounded linear operators on Hilbert spaces,
spectral theory of selfadjoint operators, applications to differential and integral
equations, additional topics as time permits.
Course Description:
Lax's book has a lot more in it than can be covered in one semester,
but it's readable and gives you a very good feeling for what the
subject is good for. I am planning to concentrate on the following topics:
- The Hahn-Banach Theorem in all its various forms (Chapters 3-4).
This subjects reoccurs in another incarnation in Chapter 8.
- Banach spaces and Hilbert spaces (Chapters 5-7). It is especially important to have
a good feel for the geometry of the latter.
- Weak compactness, a fundamental tool (Chapter 12)
- Convex sets and the Krein-Milman Theorem (Chapters 13-14)
- Uniform Boundedness and the Closed Graph Theorem (Chapter 15)
- Basics of Banach Algebras, especially commutative ones (Chapters 17-19)
- Spectral theory of self-adjoint, and especially self-adjoint
compact, operators on Hilbert space (Chapters 22-23, 28-29, 31)
If you can, try to look at the "applications" chapters, even if we don't have
time to cover them in class. They are some of the best parts of the book,
and show you the power of the subject.
Course Requirements and Grading Policy:
Homework will be assigned, collected, and graded, usually once a week.
I will probably give a take-home final exam. Grades will have the
following meaning:
- A. Did most of the problems correctly, seemed to have a
good idea of what's going on.
- B. Did a fair number of problems, but had some trouble
with them.
- C. Didn't do much of the homework, or got essentially
none of it right.
- D, F. I hope these won't arise. Basically mean "hardly
ever showed up".
It is your responsibility to turn the
homework in on time.
Course Evaluation
We are asking for your help with
on-line
course evaluation. Please provide feedback on the course and
the professor between Tue., Nov. 21 (12:00 am) and Wed., Dec. 13.
Evaluations are anonymous and will not be available to faculty
until next semester, so they cannot possibly affect your
grade.
Tentative Schedule
(Some details to be filled in later if necessary)
| Week |
Material Covered (Reading Assignment) |
Homework and Special Notes |
8/30, 9/1.
(no class 8/28) | Ch. 1, 2.1 (read for review); Ch. 3-4 |
due 9/6:
- Use the Hahn-Banach Theorem to show that if V
is a normed linear space over R (see 5.1 if you forget the definition)
and if x is non-zero in V, then there is a linear
functional f on V, of norm 1, for which
f(x) is equal to the norm of x
(and in particular is strictly positive). Deduce that
linear functionals of norm 1 separate points, i.e., that if
x and y are distinct points of V, then there is a linear
functional f on V, of norm 1, for which
f(x) is unequal to f(y).
- Suppose V
is a vector space over R with a "positive cone"; that
is, there is a subset V+ of V, closed
under addition and multiplication by non-negative scalars,
such that V = V+ - V+
and V+ and -V+ intersect only
in {0}. A linear functional f on V is called
positive if f is non-negative on V+.
(Yes, I know; "non-negative" would be more descriptive, but this
is the traditional terminology.) We write x
y if y - x lies in V+.
Suppose V has an order unit, that is, there is an
element 1 in V such that for any x in V,
there is a real number t > 0 such that tx 1. Show there is a positive linear
functional f on V, with f(1) = 1.
Next (this is harder), suppose x is a non-zero interior point in
V+. Show there is a positive linear
functional f on V, with f(1) = 1 and for which
f(x) is strictly positive.
- Here's an application of the Hahn-Banach Theorem to economics,
due basically to Debreu. Consider the vector space V =
RN, whose dimension N is the number of
goods in the economy, finite but large, and whose jth
coordinate represents the amount of the jth good.
Give this the positive cone V+
(in the sense of the exercise above)
for which the coefficients of all goods are non-negative.
We require that the amount of consumption
x is equal to the amount of production y, and both lie
in V+. (I.e., supply and demand are
balanced, and no good is produced in negative amounts.)
Assume that the production vector y must lie in
a compact convex set Y in V+
(constraining what combinations of
goods can be produced with available resources), and that the consumption
vector x must lie in a closed convex set X in
V+. (Convexity of
X is forced by "rationality of consumer choices". Can you see
why? Can you also see why it is natural to assume convexity and compactness of
Y? closedness of X?)
A price vector is a positive linear functional p on V,
assigning a non-negative numerical value (in $$) to each combination
of goods (with non-negative coefficients). Show
that if X and Y intersect, there is a choice of
x = y and a choice of a
price vector p under which the economy is in equilibrium, in
the sense that the constraints are satisfied and that the income
p(y) of the producers cannot be further increased (with
this choice of prices) without violating the constraints. (You can do
this either with or without Hahn-Banach.) Next, do a slightly harder
case. Instead of requiring x = y, require
y - x = z in V+ (production
exceeds consumption by a fixed non-negative amount, representing
"savings"). Assuming that X + z intersects Y,
show there is a choice of x, y, and p, for
which the economy is in equilibrium.
|
9/6, 9/8.
(no class 9/4) | Ch. 5, Ch. 6 |
due 9/13: Lax, Ch. 5, Exercises 5 and 6 (p. 44) and Exercise
8 (p. 48); Ch. 6, Exercises 8 and 10 (pp. 61, 62) |
| 9/11-9/15 | Ch. 6 (cont'd), Ch. 7 |
due 9/25: Exercise sheet on the Bergman kernel
|
| 9/18-9/22 | Ch. 8, Ch. 9 (selections) |
|
| 9/25-9/29 | Appendix A, the dual of C(X) |
|
| 10/2-10/6 | Ch. 10, Ch. 11 (selections) |
due 10/16: Exercise sheet on duality for Banach
spaces, weak* convergence, and convergence of Fourier series |
| 10/9-10/13 | Ch. 12, 13.1 |
|
| 10/16-10/20 | 13.2-13.4 |
due 10/25: Exercise sheet on Krein-Milman |
| 10/23-10/27 | Ch. 14 (selections) and Ch. 15 |
due 11/15: Exercise sheet on distributions, the Fourier Transform,
and linear operators |
| 10/30-11/3 | Ch. 16 (selections) and Appendix B |
|
| 11/6-11/10 | Appendix B (cont'd) and Ch. 16 (cont'd) |
|
| 11/13-11/17 | Ch. 17, Ch. 18 |
|
11/20-11/22
(no class 11/24) | Ch. 18 (cont'd) |
due 12/6: Exercise sheet on Banach
algebras |
| 11/27-12/1 | Ch. 19, Ch. 21, Ch. 22 (selections) |
|
| 12/4-12/8 | Ch. 28, Ch. 30 |
due 12/18: Final exercise sheet (cumulative) |