MAT 201B: Analysis
(Winter 2005)
MWF 1100-1150A, WELLMN 7
Course:
MAT-201B-1, Analysis
Quarter: Winter 2005
CRN: 80696
Instructor: Roman
Vershynin
(671 Kerr Hall, vershynin@math.ucdavis.edu)
TA: Momar Dieng (momar@math.ucdavis.edu)
When/Where:
Lectures MWF 1100-1150A in Wellman 7
Discussion Tuesday 1100-1150A in Wellmn 129
Office hours:
Roman Vershynin: W 2-4
Momar Dieng: TBA
TEXT
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.
ASSESSMENT
The midterm (covering Chap. 6-7) will in class on Wednesday,
February18.
Here you can find a copy of the MIDTERM and
its SOLUTIONS.
The Final Exam (comprehensive, Chapters 6-9), will be in class on
Thursday,
March 17, 8-10 a.m.
Grade distribution: Homework 25%, Midterm 25%, Final 50%.
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
Profs. Hunter
and Nachtengaele.
WEB:
http://www.math.ucdavis.edu/~vershynin/teaching/201B-2005/course.html
|
Homework 1, due
01/14 |
6.1, 6.2, 6.3,
6.4, 6.5, 6.6 |
|
Homework 2, due
01/21 |
6.7, 6.8, 6.10,
6.13, bonus: 6.14(a) |
|
Homework 3, due
01/28 |
Prove the completeness
criterion stated below the theorem on orthonormal bases: an
orthonormal set U is complete in H if and only if the set of all
(finite) linear combinations of elements from U is dense in H. |
|
Homework 4, due
02/04 |
Prove that the
Sobolev space H1(T) can be equivalently defined as the completion of
the space of continuously differentiable functions C1(T) with respect
to the H1(T)-norm. (One needs to show that C1(T) is dense in H1(T)).
|
|
Homework 5, due
02/11 |
1) Show that the
solution operator T(t) for the heat equation is the exponential of the
"second derivative operator" A, as in p.162. What is the space where
T(t) acts? |
|
Homework 6, due
02/25 |
7.17, 7.18.
Prove the claims in Examples 8.6 and 8.7 (assume A is closed), 8.10 on pp.189-190. Prove that the adjoint
operator defined by the property (8.9) is unique and is linear. Prove
that (A*)* = A and (AB)* = B*A* for all bounded linear operators A,B on
a Hilbert space. Prove the claim in Example 8.16 on p.194 |
|
Homework 7, due
03/04 |
Prove the
polarization formula (6.5) that connects an inner product (x,y) with
the norm ||x|| the inner product defines. Deduce the formula for
<y,Ax> on p.198 for positive-definite self-adjoint operators. Try
to prove that formula for general bounded linear operators A. Prove the
claims in Examples 8.31, 8.32. |
|
Homework 8, due
03/11 |
Exercises 8.16,
8.18, 9.1, 9.2, 9.3, 9.4, 9.5 ( in
9.5 assume also that A is self-adjoint) |