Homepage for Math205
Analysis II
Winter2013

Enrollment:
at the Math office.

Instructor:
Maria Schonbek
email: schonbek at ucsc dot edu
phone: 459-4657
Office: McHenry 4126
Office Hours: T: 12am -1pm, Th 2pm-3pm

Lectures: T-Th 10::-11:45, McHenry 1270

Textbook: Real and Complex Variables, 3rd edition, Rudin

Homework 10 %, Midterm 40 %, Final 50 %
No late homework accepted.

Syllabus
This Syllabus gives a general idea of the progress of the class. There will be variations depending on how fast certain topics are understood.
1. Abstract Integration: measure theory and integration.
2. Riesz representation Theorem.
3. Lebesgue measure.
4. Continuity properties of measurable functions
5. L^p spaces.
6. Complex measures : Radon-Nikodym Theorem and applications.
7. Fundamental Theorem of calculus.
8. Product measures; Fubini Theorem.

First Midterm: February 14

Homework:

Homework problems will be assigned on this webpage each  Friday  and will be due the following Thursday.
(This might be changed.) The homework has to be typed.

Homework problems:

HW1:
Page 31: 1, 2, 3, 5.

A.Let (x,M,m) be a measure space. Let A be a dense set in R. Show that the function f: X into R is measurable if an only if  the set {x in X: f(x) \geq a}
is measurable for all a in A.

B.Give an example of a nonmeasurable function.

C. Let f and g be measurable functions then
a. f+g is mesurable functions.
b.  f g is measurable functions.
c.  |f| and |f|^p are measurable functions .

HW2:
Page 32: 7, 8, 9, 10, 12.
Page 58: 1,5,6, 7.

HW3
Page 58-59:  8, 9, 15, 20
Page 58-60 : 11,12, 21, 25

HW 4
Due week after exam
Page 60: 24
Page 71: 4,  5
Page 72: 9 (warning:very hard), 10,  11, 12 (hard)

Problem 4c, Page 71
Problem 9, page 72
Problem 12, Page 72

HW 5
Page 72: 14, 16 ( show just the first part: Egorov Theorem), 20,24.
Page 132-133: 1, 2, 4, 5.
You do not need to hand this in, but you should know how to establish Prop 6.8.( page 120)

Solutions HW5
Chapter3, problem14b

HW6

Page 133: 6, 10 a,b, 13.