MATH
302: Real Analysis II
Mathematics
Department, Bryn Mawr College, Spring 2008
Professor:
Victor Donnay 
Lecture:
Mon, Wed, Fri 11 12 
Office:
Park Science Building #330 

Phone:
5265352, Email: vdonnay 
Office
Hours: Mon 13, Wed 23, Fri
1:302:30 
Prerequisites: We will
be studying material that you (might) have seen in Math 102 (Calculus 2), Math 201 (Multivariable calculus, Math
203 (Linear Algebra). We will build on the material taught in Math 301 (Real
Analysis I
Texts: Introduction to Real
Analysis, Bartle & Sherbert, Wiley 2000, 3^{rd} edition. (B)
Introductory
Functional Analysis with Applications, E. Kreyszig, Wiley Classics, 1989. (K)
Course Web Site: accessible from Prof. Donnay's homepage
All materials for the course will be found on
the web site or at the course Blackboard site.
TA:
Sherry Teti (steti@brynmawr.edu) will be the TA for the
course and will run several help sessions per week:
Mondays
3 pm to 5 pm in Park 337; Tuesdays
3:30 pm to 4:30 pm in Park 349; Thursdays 1 pm to 2
pm in Park 349
Goals of the Course: In this
course, you will:
Communicate your mathematical
reasoning in writing and verbally, both via informal arguments and via more
formal proofs.
Develop your ability to work
as an independent and selfsufficient learner:
What
to do when you do not know what to do
How
to take what you have learned in one situation and apply it to a new and
different situation (transfer of knowledge)
Get
comfortable with not knowing the answer immediately
Learn
material we have not covered in class by reading the book and applying this
newly learned information to solve problems.
Decide
for yourself whether you understand material and learn how to ask yourself
questions to check your understanding.
Become part of a community of learners who
support, encourage and learn from one another.
You will demonstrate your progress in these areas by
undertaking a final project on a topic of your own choosing.
Topics:
Metric spaces (Ch. 11 B, Ch. 1 K). We will
extend the notions of analysis that you have learned for R (sequences, limits,
continuity) to the more general setting of metric spaces.
Sequences of functions: (Ch. 8 B).
Series (Ch. 3.7 K, Ch. 9 K).
Differentiation (Ch. 6 B)
Integration (Ch. 7 B)
Normed spaces, Banach Spaces (Ch 2.14, K)
Inner Product Spaces, Hilbert Spaces (Ch.
3.16).
Computer Assignments: We will have occasional computer assignments and will
sometimes use computers during the course. We will use Mathematica; but no
previous experience is assumed.
Exams:
There will be a midterm exam, a final exam
(both take home exams) and a final project. The tentative schedule
for the exams is:
1st
exam: probably in the 6^{th}
week (Feb 25 Feb 29).
2nd
exam: probably in the 12^{th }week
(April 14 18).
Final Project: Due
during exam period.
Students will work in two person teams on a
project of their choosing. The project might involve using material from the
course to study an applied situation, examining a theoretical issue in more
depth or studying a topic that extends the material from the course. Projects
will be written up in the form of a paper (10  15 pages). During the last two
weeks of the term, teams will give short (10 – 15 minute) presentations
about their projects to the class (providing the class does not get too big!).
Homework:
Homework will be assigned each week. The homework related to the
Monday – Friday classes will be due the following Wednesday. Late work will not be accepted unless there is a special situation (ex. serious
medical problem) and you get my permission ahead of time.
The best way to learn mathematics is by
doing. At this level of more theoretical mathematics, problems can take a lot
of thought and experimentation to complete. Part of the goal of the course is
to help you develop strategies to attack these hard problems (draw pictures,
make simpler miniproblems, read the text very carefully, discuss with your
classmates).
Much learning happens by trying, doing as
much and as well as you can, then getting feedback and trying again. So there
will be some HW problems where you will be asked to revise and resubmit.
Quizzes: We may have occasional miniquizzes to give you and me
a chance to gauge your understanding of key concepts in the course.
Classroom:
During class, there will be a mixture of
lecturing by the professor and time spent by the students working out problems,
discussing their results in groups and having whole class discussions. Research
has shown that this type of active participation leads to improved learning.
The group work does not go well when members
of the group are absent. Therefore it is important that you attend to class.
Please be respectful of your fellow students.
If you decide to take this course, you must commit
to attending class regularly. Attendance will be taken and substandard
attendance will be taken into account in deciding your grade.
Final grades will be determined using the
following percentages:
Homework, quizzes, class participation 
25% 
Test 1 
25% 
Test 2 
25% 
Final Project 
25% 
Total 
100% 