MCS 331: Real Analysis
Fall 2001
College catalog's course description
An introduction to the techniques and theorems of real analysis.
Topics will include: Metric spaces and Real function theory, including
Riemann-Stieltjes integration and sequences and series of functions.
Prerequisite: MCS 220 (Introduction to Analysis)
Real course description
Traditionally courses in real analysis present the content in a
polished form. The definitions are stated precisely, and the best
possible theorems are set forth and proved via terse and flawless logic.
The presentation conceals the tracks followed by generations of
mathematicians, who struggled for decades to find the "right"
formulations of concepts that make the subsequent
theory look nice, to obtain the neatest statements of theorems, to write
the slickest proofs, and to arrange all the theorems in proper logical
order.
This course, in contrast, will trace some of the historical development
of analysis, revealing some of the confusion and challenges the
founders of modern analysis had to overcome,
in the hope that this will lead students to a better conceptual
understanding of, and appreciation for, the modern rigorous approach
to real analysis.
According to Bressoud,
the mathematical crisis struck in 1807,
when Fourier submitted his famous manuscript on the
theory of heat (Bressoud 1). The issues raised by the introduction of
Fourier series at that time exposed weaknesses in the foundations of
calculus. "[T]hese infinite summations cast doubt on what scientists
thought they knew about the nature of functions, about continuity, about
differentiability and integrability. If Fourier series were to be
accepted, then all of calculus needed to be rethought." (8)
This course will carry out that rethinking, and will set the student on
the path to further abstraction and generalization. As Sprecher notes,
"A student's training in calculus is generally related to his [sic]
everyday experiences, and the symbols he manipulates represent
familiar objects. The essence of mathematics, however, is its capacity
for abstractions and generalizations. ... Many abstractions and
generalizations originate with the real line, which is used as the
unifying theme of the text." (Sprecher vii)
Much of the mathematical development in this course will be devoted to
answering a couple of big questions.
-
BIG QUESTION #1:
What is the structure of the real number system?
In particular, "How are the real numbers different from the Rational
Numbers?" (Schramm 1)
-
BIG QUESTION #2:
What is the nature of functions of a real variable?
In particular, which functions can be represented by infinite series,
especially Taylor and Fourier series, and how?
| Week
| Class Dates
| Chapters
| Topics
| Tests
|
|
|
| B=Bressoud
S=Sprecher
|
|
|
| 1
| 9/5-7
| B 1
| Crisis in mathematics
|
|
| 2
| 9/10-14
| B 2
| Infinite summations
|
|
| 3-4
| 9/17-24
| S ch. 3
| The real number system
|
|
| 4-5
| 9/26-10/5
| S ch. 5
| The structure of point sets
|
|
| 6
|
|
|
| M 10/8
|
| 6-7
| 10/10-17
| B 3
| Differentiability & continuity
|
|
Spring
Break
| 10/19-10/22
|
|
|
|
| 8
| 10/24
| B 3.5
| Consequences of continuity
|
|
| 8
| 10/26
| S ch. 6
| Continuity
|
|
| 9
| 10/29
| S ch. 7
| Differentiability
|
|
| 9-10
| 10/31-11/9
| B 4
| The convergence of infinite series
|
|
| 11
| 12
| S ch. 4
| Sequences and series of numbers
| W 11/14
|
| 11-12
| 11/16-11/21
| B 5
| Understanding infinite series
|
|
| 12
| 11/22-25
|
| Thanksgiving break
|
|
| 13
| 11/26-30
| B 6-7
| Return to Fourier series
Integration
|
|
| 14-15
| 12/3-12
| S 8
| Spaces of continuous functions
Wrap-up
|
|
| 15
| 12/14
| Friday
| Final exam
| 8:00-10:00
|
This is a tentative syllabus. Changes may be made during the semester.
Instructor: John Holte
Office: Olin 307
Telephone: x7465
Office hours: M 1:30-2:20, T 9:00-9:50, W 1:30-2:20,
F 9:00-9:50
Feel free to see me at other times as well.
E-mail: holte@gac.edu
My homepage: http://www.gac.edu/~holte/
Course web site
Textbooks
Plan to read the texts carefully before and after each class.
In class we will discuss the readings or consider problems or proofs
related to the readings.
Although the number of pages we cover per
week may be more than in an average math course, you should not find the
amount of reading uncomfortable, because the texts cover themes you
have seen several times in previous courses.
Class meetings
8:00-8:50 Monday, Wednesday, and Friday
Classroom: Olin 319
Regular attendance, both physical and mental, is required.
Class meetings will be used for a variety of activities: lectures,
discussions, and informal or formal presentations of solutions or
proofs.
- Acknowledge your sources (people and texts).
- In nontrivial problems, show how you get your answers.
- Turn in neat, well-written solutions, not messy first drafts.
Trim "fringes." Staple multiple-page submissions.
- Do not copy collaborative solutions; write up solutions in your own words.
- Turn in homework on time. Each class day late reduces the possible points
by 25%.
- Do extra credit problems entirely on your own.
Academic honesty
Page 68 of the Gustavus Guide
states in part: "The faculty of Gustavus Adolphus College expects all
students to adhere to the highest standards of academic honesty... In
all academic exercises, examinations, papers, and reports, students
shall submit their own work. Footnotes or some other acceptable form of
citation must accompany any use of another's words or ideas."
A first case of cheating will result in a zero on the test or
assignment in question.
A second case will result in a failing grade for the course.
Exams
There will be two unit exams and one comprehensive final examination.
Some or all of the unit exams may be take-home examinations due on the
class day following the day it is given out.
See the syllabus for the exam schedule.
Make-up policy
Make-up exams will not be given except for medical or family emergency
reasons.
Students who will be absent from an exam for a school-sponsored event
should arrange with me in advance to have an exam sent along with the coach.
If you miss a class for any reason, you are still responsible for
learning what was covered in class and for getting any homework that is
due turned in on time. If there is a compelling reason for granting an
extension, you must negotiate the arrangements with me.
Late homework will be penalized as stated under
Homework rules.
Grading
| Fraction
| Work
|
| 1/3
| Problem assignments
|
| 1/6
| Exam 1
|
| 1/6
| Exam 2
|
| 1/6
| Final exam
|
| 1/6
| Participation & formal presentations
|
Extra help
-
I encourage you to see me during my office hours and at other
times when my office door is open.
-
If you have a disability that requires an accommodation, please
see me privately as soon as possible.