MATH 6700: REAL ANALYSIS
Schedule for Week 15. Additional Office Hours on
Monday, December 14, from 10 a.m. - 1 p.m.
Course Info
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Weekly
Schedules
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Course
description: This is a first part of two-semester course on real
analysis. This semester we study theory of functions of real variables
including Lebesgue measure and Lebesgue integral, differentiation and
integration. We also study some elements of functional analysis:
classical Banach spaces, general metric and topological spaces.
A list of problems to work
will be placed
at
the course webpage (see homework problems).
Your homework will be collected and graded on regular basis. It is
recommended
to keep your homework
to
use in preparation for mid-term and final exams.
Textbook:
Real Analysis (Third edition) by H.L.Royden, Prentice
Hall.
Recommended complementary reading is Counterexamples
in Analysis (Dover Books on Mathematics) by Bernard
R. Gelbaum,
John M.
H. Olmsted. See it at amazon.com.
Office Hours: Tuesday 9:00-9:50 a.m.,
Thursday: 9:00-11:00 and Friday 9:00-9:50 a.m. in
Everett
Tower 5527. See a
detailed week
schedule .
Week 1:
September
7 - September 11
Topics for
reading
- Sections
1.1-1.6 Elements of set theory
- Sections
1.7-1.8 Relations, partial orderings and Hausdorff maximal principle
Homework #1
(due to
Wednesday,
September 16)
Week
2: September 14 - September 18
Topics for
reading
- Sections
1.7-1.8 Relations, partial orderings and Hausdorff maximal principle
- Sections 2.1
-2.2 Axioms for the real numbers, natural and rational numbers
- Sections 2.3
- 2.4 Sequences of real numbers
- Section 2.5
Open and closed sets of real numbers
Homework #1
(due to Wednesday,
September 16). Set of
bonus problems for
homework # 1 due to September 18.
Week
3: September 21 - September 25
Topics for reading
- Section 2.5
Open and closed sets of real numbers
- Section 2.6
Continuous and semicontinuous functions
- Section 2.7
Borel sets
Homework #2
(due to Friday,
September 25). Bonus problems #2
(due to Friday, Septembe 25)
Week
4: September 28 - October 2
Topics for
reading
- Section 2.7
Borel sets
- Sections 3.1
- 3.2 Outer measure
- Section 3.3
Measurable sets and Lebesgue measure
Homework
#3 (due
to Friday, October 2 ).
Week
5: October 5 - October 9
Topics for
reading
- Section 3.3
Measurable sets and Lebesgue measure
- Section 3.5
Measurable functions
- Section 3.6
Littlewood's three principles
- Section 4.1 Riemann integral
Homework
#4 (due
to Monday,October 12).
Week
6: October 12 - October 16
Topics for
reading
- Section 3.6
Littlewood's three principles
- Section 4.1 Riemann integral
- Section 4.2
Lebesgue integrasl of bounded functions
Week
7: October 19 - October 23
Topics for
reading
- Section 4.2
Lebesgue integrasl of bounded functions
First midterm exam
on October 19, Monday: Sections 1.1 - 3.6. Homework #5
is due
on Wednesday, October 28.
Week
8: October 26 - October 30
Topics for
reading
- Section 4.3
Integrals of positive functions (Fatou's lemma, Monotone Convergence
Theorem, absolute continuity of integrals)
- Section
4.4 General Lebesgue integral
Week
9: November 2 - November 6
Topics for
reading
- Section
4.4 General Lebesgue integral
: problems
- Section
4.5 Convergence in measure
- Problem
solving
session
Homework #5
is due
on Friday, November 6.
Week
10: November 9 - November 13
Topics for
reading
- Section 5.2 Functions of bounded variations
- Section 5.3 Differentiation of
an integral
- Section 5.4 Absolutely
continuous
functions
Homework
#6 is due to Monday, November 16
Week
11:
November 16 - November 20
Topics for
reading
- Section 5.3 Differentiation of
an integral
- Section 5.4 Absolutely
continuous
functions
- Section 5.5 Convex Functions
Homework
#7 is due to Wednesday, November 25.
Second
midterm exam on Monday,
November 23. Problems and results from Chapter 4 (including convergence
in
measure).
Week
12: November 23 - November 27
Topics for study
and reading
- Section 5.5 Convex Functions
- Section
6.1 L^p spaces
Homework
#7 is due to Wednesday, November 25.
Second
midterm exam on Monday,
November 23. Problems and results from Chapter 4 (including convergence
in
measure).
Week
13:
November 30 - December 4
Topics for
study and reading :
- Section
6.1 L^p spaces
- Section
6.2 Minkowski and Holder
inequalities
- Section
6.3 Convergence and
completeness
Week
14: December 07 - December 11
Topics for
study and reading :
- Section
6.5 Bounded linear functionals in L^p spaces
- Review of the course
Last homework is due
to Wednesday, December 9
Homework
Problems
Note that problems
marked with
*
are extra.
Homework #1
(due to Wednesday, September 16)
Section 1.1 (p.8) #5
Section 1.2 (p.12) #6
Section 1.3 (p.16) #9, 10, 11, 12 b,c ,e, 13, 15, 16b,c
Section 1.4 (p.19) #19a,b
Section 1.6 (p.22) #22, 24, 25
Section 1.8 (p.26) #30
Homework
#2 (due to Wednesday,
September 23)
Section
2.1 (p.34) # 1, 4 b), c), 5a)-e)
Section 2.4
(p.38) #7, 8, 13, 14, 16, 22
Section 2.5 (p.46) #24, 25, 26, 27, 31, 34
Homework #3 (due
to Friday, October 2, )
Section 2.6 (p. 49 ) # 41,
42, 43, 46, 48, 50 a, c, e
Section 3.2 (p. 58) # 6, 7, 8
Section 3.3 (p. 64) # 9,
10.
Homework
#4 (due
to Monday, October 12)
Section 3.5 (p.70) # 20, 21, 24, 25, 26
Section 3.6 (p.73) # 29, 30
Homework
#5
(due to Friday,
November 6)
Section
4.1 (p.76) #1 a
Section 4.2 (p.89) # 3, 5, 7, 8
Section
4.4 (p. 93) # 10 a, b, 14 b, 15 a,b, 16, 17, 18
Homework
#6
(due to Friday,
November 13)
Section
5.1 (p.101) #1, 2 a,b, 3 a, b, 6 a
Section 5.2 (p.104) # 7a, 8 a,b, 9, 10 (bonus problem), 11
Homework
#7 (due to Wednesday, November 25)
Section
5.4 (p.110) #12, 13, 14 a,b, 16a (bonus) , 20 a, b
Section 5.5 (p.116) # 23
a,b,c, 25 a,b, 27, 28
Homework
#8 (due to Wednesday, December 9)
Section
6.3 (p.126) #10, 17, 18
Section 6.5 (p.134) #21 a, 24
(bonus)
Exams
- First mid-term exam : Monday,
October 19
- Second mid-term exam: Monday,
November 23.
- Final exam (comprehensive) : Thursday,
December 17 at 10:15 - 12:15.
Grades
and Makeup
Policy
Grades will be
assigned using a sliding
scale.
The usual passing level "C" is between 65% and 68%. Someone with 88%
may
get "A". After each test we'll get an approximate passing level. Do not
panic
if your grade seems low, come and talk with me about your
perfomance.Various
numbers below determine the relative weights of exams' and
quizzes' grades: Total number of points 500. The final exam - 200
points. Each midterm
exam - 80 points, homework grade - 140.
In general,
there will be NO makeups
for
exams. If you miss an exam for a valid well-documented reason then
arrangements
may be made on an individual basis.
Academic
Integrity
You
are
responsible
for making yourself aware of and understanding the policies and
procedures
in the Graduate Catalog
(pp. 26-28)
that
pertain to Academic Integrity. These policiesinclude cheating,
fabrication,
falsification and forgery, multiple submission, plagiarism, complicity
and
computer misuse. If there is reason to believe you have been involved
in
academic dishonesty, you will be referred to the Office of Student
Conduct.
You will be given the opportunity to review the charge(s). If you
believe
you are not responsible, you will have the opportunity for a hearing.
You
should consult with me if you are uncertain about an issue of academic
honesty
prior to the submission of an assignment or test.
Comments,
questions,
problems to ledyaev@wmich.edu
.
Last modified
:December 4 , 2009
