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: 526-5352, E-mail: vdonnay	Office Hours: Mon 1-3, Wed 2-3, Fri 1:30-2:30

Pre-requisites: We will be studying material that you (might) have seen in Math 102 (Calculus 2), Math 201 (Multi-variable 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 self-sufficient 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 2.1-4, K)

Inner Product Spaces, Hilbert Spaces (Ch. 3.1-6).

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 mid-term 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 mini-problems, 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 mini-quizzes 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%