**Real Analysis and Linear Spaces I - Math 731 - Fall 2007**

**Lecturer:**- Irina Mitrea, 225 Kerchof Hall, im3p(at)virginia(dot)edu, Phone: 982-2787.

**General description:***Topics include***functions**(semicontinuous, continuous, uniformly continuous, absolutely continuous, of bounded variation, differentiable functions; extreme value, intermediate value, mean value theorems; Series of functions, power series, and power series of elementary functions, uniform convergence, Weierstrass M test),**Lebesgue measure and Lebesgue measurable functions**(outer measure, Cantor Sets, characterizations of measurability, nonmeasurable sets, convergence in measure,Egorov's and Lusin's theorems)**the Lebesgue integral**(integration of measurable functions, monotone and dominated convergence, Fubini's and Tonelli's theorems),**differentiation**(Lebesgue's differentiation theorem, the Vitali covering lemma, differentiation of monotone functions)**Lebesgue spaces of p integrable functions**(Holder's and Minkowski's inequalities, Banach and metric space properties, the space of square integrable functions, orthogonality, Fourier series, Parseval's formula, Hilbert spaces).

**Textbook:**-
Richard L. Wheeden and Antoni Zygmund,
**Measure and Integral, An introduction to Real Analysis**.

**Recommended References:**-
R.G. Bartle,
**Elements of Real Analysis**. -
W. Rudin,
**Principles of Mathematical Analysis** -
H.L. Royden,
**Real Analysis** -
G. Folland,
**Real Analysis**

**Course web page:**-
http://people.virginia.edu/~im3p/mathpage731.html

**Examinations:**- Prelim 1: Friday, September 21st, 13:00 - 13:50.
- Prelim 2: Monday, October 22nd, 13:00 - 13:50.
- Prelim 3: Monday, November 19th, 13:00 - 13:50.
- Final Exam: Monday, December 10th, 9:00 am - 12:00 noon.
*Notes, textbooks and calculators are not allowed during prelims and the final examination.*

**Lectures:**

** Office Hours:**

**Grading:***Prelims count 20% each, the final 20%, and the remaining 20% is apportioned among homework, quizzes and class participation.**A: 90-100**B: 80-89**C: 70-79**D: 60-69**F: <60*

**Quizzes:***There will be a short quiz every Friday, unless otherwise announced. It will be returned to you on Monday.*

**Homework**:*To be handed in during lecture meeting. Do all problems assigned. These will be graded and returned to you the next week. Please be sure that your work is legible.***No late homework will be accepted.**You may work cooperatively on assignments provided- You write up the solution yourself.
- You put a note on your homework indicating the names of anyone you worked with.

**Incomplete**:*An incomplete will be given only in those rare circumstances where a student has completed all but a small portion of the course with a grade C or better and a severe, unexpected event prevents him/her from completing the course. In particular, if you get behind in the course you cannot ``bail out" by taking an incomplete.*

**Attendance and absences**:*You are responsible for the material covered in class, whether you attend or not. You are also responsible for the announcements made during class; these may include changes in the syllabus.*

**Academic honesty***is fundamental to the activities and principles of a University. Any effort to gain an advantage not given to all students is dishonest whether or not the effort is successful. When in doubt about plagiarism or collaboration, consult the course instructor. The academic community regards academic dishonesty as an extremely serious matter, with serious consequences that range from probation to expulsion.*

**If you need accommodations because of a disability**,*if you have emergency medical information to share with me, or if you need special arrangements in case the building must be evacuated, please inform me immediately. Please see me after class, or at my office.*

**Words of advice:***I encourage you to ask questions about the HW.***Keep up with the latest materials covered**; experience shows that, otherwise, you are likely to get poor grades on the exams. If you miss a class, make sure you get the notes from someone else who attended it. The professor will not assist any absentee to find out what happened in his/her absence. Hard work and regular attendance will get you through this course.