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,
TA: Momar Dieng (
    Lectures MWF 1100-1150A in Wellman 7
    Discussion Tuesday 1100-1150A in Wellmn 129
Office hours:

    Roman Vershynin: W 2-4
    Momar Dieng: TBA

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.

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.


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.
6.12, 6.11 assuming H is separable, 7.1

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)).
7.2, 7.3, 7.4, 7.5

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?
2) Prove that the operators T(t) form a Co-semigroup (prove the properties listed on p.163)
7.6, 7.7, 7.8

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.
Exercises 8.1, 8.2, 8.3, 8.6, 8.12

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)