Analysis
I, Math 4317
Lectures: MWF 12-1 pm in Skiles 154
Instructor: Michael Loss (loss@math.gatech.edu)
Office: Skiles 214 B
Office hours: W, F 1-2 or by appointment
This page will be frequently
updated,
so, please, consult it often.
Analysis, it is
sometimes said, is the art of taking limits! Many interesting
mathematical objects arise as limits and one is faced with the
challenge of calculating
with them. The square root of two, the elementary transcendental
functions are elementary examples, however,
solutions of partial
differential equations, most of the time, appear as limits as
well. It is therefore important to understand how to compute
with limits. Analysis
provides a number of notions and tools for dealing with limits and
this course is a gentle introduction to them. Since
the notions in
analysis are subtle, it is sometimes difficult to get an
intuitive feeling of what these things mean. The only way to stay
on firm
ground in this field is by
understanding and doing proofs. You have to be able to prove
certain statements, i.e., to either derive a statement
through logical steps from
another one, or to find a counterexample. Thus, this course is not
so much about solving particular problems
but to teach you the
language of analysis.
The text for the course
is `Introduction to Analysis' (IA) by Maxwell Rosenlicht,
Dover 1986, ISBN
0-486-65038-3. Some additional notes you will find on this link.
Among the topics to
discuss are elementary set theory, functions, the real
numbers,
metric spaces, open and
closed sets, compactness, continuous functions as well as
differentiation and integration.
Practice is important.
You will have to do homework that is regularly graded. In addition
there will be two tests and a final exam.
Your grade will be based on
how well you solve the homework and the tests. The lowest test
score will be dropped. Homework
counts 30%, the test counts
30% and the final counts 40% towards your grade. The grade scheme
is 90% or more A, 80% or
more B, 60% or more C,
50% or more D and below 50% F.
Here are some
Practice Tests
Now a few words about the
homework. Every week I assign problems and half of them will be
graded, but I am not telling you
which ones. It goes without
saying that you to do the homework yourself, i.e., you do not copy
from others. I do not mind that,
should you get stuck, you
ask your fellow students for help, but you have to try first
yourself. Let me also remind you to uphold
the honor code at all times.
You find more information at http://www.honor.gatech.edu/plugins/content/index.php?id=9
Please turn in the home work
always on Wednesdays, the week after it was assigned, in class.
Here is a tentative
syllabus. I do not want to stick to it in a rigid fashion.
There may be a topic that
for some reason will require that we spend more time
on it. I would like to have
this option. It is better to understand fewer topics
very well than lots of
topics badly.
Please note that weeks 3-14 have
an exercise group assigned to the homework.
Each group solves the
homework in detail and submits it to me on Friday of the
following week in class.
This group - solutions with my corrections will be posted
on the web. We will organize
this at the beginning of the course and I will post the
groups as well as their
meeting times on this page.
The
Final Exam is on Dec 14 (Fri) 11:30am - 2:20pm in
Skiles Room 154
Week 1: August 21
- 24. Assigned Text: Chapter 1 in IA
Homework: Do all the
problems at the end of Chapter 1 in IA.
Week 2: August 27 -
31. Assigned Text: Chapter 2 in IA
Homework: Do
problems 2, 4, 6, 7, 9, 10, 11, 12, 13, 16 on pages 29-31
Exercise
group: Keane, Ali, Choi. Possible meeting
time: Mo 1-2, Wed 1-2 Due date September 7.
Homework 2 Solutions
Comments on the solutions
Week 3: September 3 -
7. (September 3 is Labor Day) Assigned Text: Chapter 3,
Section 1- 3, in IA
Homework: Do problems
1, 2, 5, 6, 7, 10, 11 on pages 61-62 of IA.
Exercise
group: Xu, Mohammadi. Possible meeting
time: Tu 1-3 Due date September 14
Homework 3 Solutions
Comments on the solutions
Week 4: September 10
- 14. Assigned Text: Chapter 3, Section 4 - 6, in IA
Homework: Do problems
12, 13, 15, 18, 20, 22 on pages 62-64 of IA.
Exercise
group: Woolfitt, Siegel, Nguyen. Possible meeting
time: Wed 2-4 Due date September 21
Homework 4 Solutions
Week 5: September 17
- 21. Assigned Text: Chapter 3, Section 4-6, in IA
Homework: Do problems
19, 21, 27, 29, 31, 34, 35 on pages 62-65 in IA.
Exercise
group: Bakir, Loupos, Yan. Possible meeting
time: Tu 3-5 Due date September October 3
Homework 5 Solutions
Week 6: September 24
- 28. Assigned Text: Finish Chapter 3.
There will be a
50 minute test in class on Friday September 28 on the topics
covered
in the week 1-5
There
will be a review session on Monday September 24 from 6-8pm in
Skiles room 154.
Homework 6 Solutions
Solutions of Test 1
Week 7:
October 1 - 5. Assigned Text: Chapter 4, Section 1-3
in IA
Homework: 1a), 1d),
2, 4, 9, 10, 11on page 91, 92 of IA.
Exercise group: McEvoy, Hombostel,
Cheung. Possible meeting time:
Th 11-1 Due date October 12
Homework 7 Solutions
Week 8:
October 8 -12. (October 12 at 4:00 pm is the deadline to
drop an individual course with a grade of ``W'')
Assigned Text:
Chapter 4, Section 1-3 in IA
Homework: 1b), 1c),
11, 14, 15, 16 on page 90-92 in IA.
Exercise
group: Parker, Hicks, Bradshaw.
Possible meeting time: Mo 10-11, Wed 10-11 Due date October
19
Homework 8 Solutions
Week 9:
October 15 - 19. (October 13 - 16 Student recess)
Assigned Text:
Chapter 4, Section 4-6 in IA
Homework: 17-21, 23
on pages 92/93 in IA.
Exercise group: Lee,
Garvan, Sadlon. Possible
meeting time:Wed 2-3, Fri 2-3 Due date October 26
Homework 9 Solutions
Week 10:
October 22 - 26. Assigned Text: Chapter 4-6 in IA
Homework: 32, 34, 35,
40, 42 on pp 94/95 in IA.
Exercise group: Goel,
Issac. Possible meeting time:Fri
2-3, Fri 4-5 Due date November 2
Homework 10 Solutions
Week 11:
October 29 - November 2. Assigned Text: Chapter 5 in
IA
There will be a
50 minute test in class on Wednesday October 31 on the topics
covered
in the week 1-10.
Solutions of Test 2
Notes on integration
Week 12:
November 5 - 9. Assigned Text: Chapter 6 Sections 1, 2 in IA
Homework: Problem 5,
6, 8 on page 109 in IA, 2, 3, 7 on page 132/133 in IA
Exercise group:
Rogers, Volgas, Diaz-Mercado. Possible meeting
time:Mo11-12, Wed 11-12 Due date November 14
Homework 12 Solutions
Week 13:
November 12 - 16. Assigned Text: Chapter 6 Sections 1-3 in
IA
Homework: Problems
16, 17, 21, 23, 24 on pp 133/134 in IA.
Exercise
group: McKay, Wells, Walker. Possible meeting time: Tu
10-11, Th 10-11 Due date November 21
Homework 13 Solutions
Week 14:
November 19 -23. Assigned Text: Chapter 6 Section in
IA
Homework: Problems 9, 11,
12, 13, 15 on page 161/162 in IA.
Exercise group:
Chen, Crews, Belcher. Possible meeting
time: Tu 11-1 Due date November 28
Notes on equicontinuity
Homework 14 Solutions
Week 15:
November 26 - November 30. Assigned Text: Chapter VII in IA
Homework problems: 39
and 40 on page 166 in IA.
Exercise
group: Vo, Kim, Moore.
Possible meeting time: Wed 3-4, Fri 3-4 Due date December 5.
Homework 15 Solutions
Week 16:
December 3 - 7. Assigned Text: Continue Chapter 8 Section 3
in IA
The Weierstrass approximation
theorem