MAT 201C: Analysis (Spring 2005)


MWF 1100-1150A, WELLMN 7

Course: MAT-201C, Analysis
Quarter: Spring 2005
Instructor: Roman Vershynin (671 Kerr,
Zeke Kwan Vogler (474 Kerr,
    Lectures MWF 1100-1150A in Wellman 7
R 0210-0300P in HICKEY GYM 290
Office hours:
    Roman Vershynin: F 1:30-3
Zeke Kwan Vogler: Tu 2-3

Applied Analysis,
by John Hunter and Bruno Nachtergaele, World Scientific 2001 ISBN 9810241917.
You can view the text chapter by chapter from Prof. Hunter's webpage in postscript and pdf formats.

Grade distribution: Homework 25%, Midterm 25%, Final 50%.

The Midterm Exam will be in class on Friday 05/13. Solutions to the Midterm
The Final Exam (comprehensive, Chapters 10-11), is on Saturday, June 11, 10:30am-12:30pm, Wellman 7. Solutions to the Final

Most homework assignments will be from the Hunter-Nachtengaele's textbook. They will be posted on this webpage. The assignments will be due at the start of class. No late homeworks will be accepted. If you miss a homework for a medical reason, that homework will not count towards the final grade and you will not be required to submit that homework later. Keys or solutions to the homeworks will also be posted on this webpage.

There will be no makeup midterms given. If you miss the midterm for a medical reason, the final exam will count for 75% and the midterm will not count.

OLD MATERIALS can be found in the webpages of Prof. Nachtengaele.


Homework 1, due 04/11

10.1, 10.2, 10.3, 10.4, 10.5, 10.6


Homework 2, due 04/18

10.7, 10.8, 10.10, 10.11 (Exercise 10.9 is not assigned anymore)

Homework 3, due 04/25

1. Prove that the operator A3 in Example 10.1 is closed.

2. Prove that if A is a closed linear operator then the resolvent R_lambda is bounded (once it is well defined).

3. Refer to Example 10.10; prove that Neumann, mixed and periodic boundary conditions are self-adjoint.
Solve the Dirichlet boundary value problem for the following ODE:
u'' + u = f,
u(0) = u(1) = 0
where f is a square integrable function.

4. An alternative way to define weak derivatives is as the L2 limit of smooth derivatives. Thus we say that u is weakly differentiable and its weak derivative is u'=v if there is a sequence of smooth functions u_n such that u_n converges to u and v'_n converges to v in L2. Prove that this definition of weak derivatives is equivalent to the Sobolev space definition.

Homework 4, due 05/02

1. Prove that a function u from L2 on the real line belongs to H2 and has a second weak derivative v = u'' if and only if the identity
<u, h''> = <v, h> holds for all test funcitons h (i.e. infinitely smooth and of compact support).

2. Prove a similar statement for functions on the interval (0,1).

3. Give a detailed proof of Example 10.28 on p.267.

Exercises 10.13, 10.14, 10.15, 10.16

Homework 5, due 05/09

1. Prove that (11.3) defines a seminorm for each alpha and beta.

2. Excercise 11.1

3. Prove the claim below Definition 11.2 about the topology on X obtained from a family of seminorms. Namely, prove that a sequence (x_n) converges to x if and only if it converges with respect to every seminorm, i.e. p_alpha (x_n - x) converges to zero for all alpha.

4. Exercise 11.3

Homework 6, due 05/23 Note: Solutions to the midterm are posted.
11.5, 11.6, 11.7, 11.8, 11.9, 11.10
Homework 7, due 06/01 11.11, 11.12 (see Definition 11.38).
Find the orthonormal set of eigenvectors (and eigenvalues) of the Fourier transform L2 -> L2. (see p.312). Prove your claims.
Example 11.40 (only the first part, on the differential operator -Delta + I).
11.13, 11.14, 11.15, 11.16
Homework 8, due 06/08 11.17, 11.18, 11.19, 11.20, 11.21, 11.27, 11.28 (Use Example 11.33). NOTE: Please check errata