Instructor: Jonathan Sands, Room 404, 16 Colchester Ave. homepage: http://www.cems.uvm.edu/~sands/
Reach me at: 656-4339 or sands@cems.uvm.edu
Office hours: Mon., Wed. and Fri. 11:00-12:00, Tues. 1:00-2:00 I will usually also be in my office around mid-afternoon
if you wish to drop by or make an appointment.
Course website: http://www.cems.uvm.edu/~sands/333/index.html
Goals:
What should be meant by the measure of a set? How can integration be
defined so as to apply to a larger class of functions than the Riemann
integrable ones? What are some of the most powerful theorems in the
theory of integration? How can we view certain important classes of
functions as normed vector spaces, and what theorems can we prove as a
result? How can measures and linear functionals be represented by
integration? We aim to answer these questions through a mastery of the
fundamentals of measure theory and the theory of the Lebesgue integral,
followed by a brief introduction to Lp spaces and representation theorems.
Prerequisites: Students should have a solid knowledge of introductory real analysis including continuity, sequences, series, convergence, connectedness, compactness, and completeness. I will help point out important concepts to review along the way.
Text: Principles of Real Analysis, Third Edition, by Aliprantis and Burkinshaw. We will also use Counterexamples in Analysis by Gelbaum and Olmsted as a supplementary resource. Standard references which may be useful for supplementary reading are: Real Analysis by Royden, Measure Theory by Halmos, Real and Complex Analysis by Rudin, and Real and Abstract Analysis by Hewitt and Stromberg. Other textbooks on the subject may be found in the library. It can be beneficial to find a supplementary text that suits your individual taste or just provides you with alternative explanations of the material.
Homework: Homework assignments will be posted on the homework webpage, and will normally be due on Tuesdays at the beginning of class. Informal consultation between students is encouraged, but the details of writing up solutions must be done individually. All homework should be presented clearly and completely. Late homework can only be accepted by special prior arrangement.
Mid-Term Exam: There will be a 2-hour closed-book honor-system take-home exam during the last week of October.
Class Presentations: Each student will make at least two class presentations. A presentation could be on a specific topic that fits in with the course material, or a solution to a problem of special interest. You are welcome to propose presentations that you would like to make, otherwise they will be assigned! Our supplementary resource, Counterexamples in Analysis, can be a good source of ideas for presentations.
Oral Final Exam: The final exam will be an oral exam scheduled individually with the instructor. Topics will be specified in advance. The idea is to give students some experience which may be valuable in preparing for Oral Comprehensive Exams.
Course Grades:These will be determined by weighting the different course requirements as follows. Homework: 50%, Class Presentations: 10%, Mid-term Exam: 20%, Oral Final Exam: 20%. Mastery of the fundamentals is the key consideration.
Expectations: The UVM Academic Honesty Policy and Classroom Code of Conduct is in effect, as always. In particular, always be sure to give proper attribution for work or ideas that are not your own. All coursework should be presented clearly and completely. Students must respect the thoughts and ideas of both the instructor and the other students. Such respect calls for good attendance, promptness, and making every effort to avoid interrupting class time.
Special Needs: If you are eligible and need an accomodation, please inform me and provide appropriate documentation during the first two weeks of class so that this can be implemented.
Missed work: Make-up tests can be arranged in the case of an emergency, if you inform me before the start of the test. UVM policy also allows students to make up work that conflicts with documented religious observances or intercollegiate athletic events. Meet with me to discuss this during the first two weeks of the semester.
| Created by Wolfram Mathematica 6.0 (21 August 2007) | |