Real Analysis I : Math 623

Meeting : M W 8:40--9:55    LGRT 1234
The class does not regularly meet on friday. However the class will meet a few times on fridays to compensate some of my absence during the semester.

Instructor : Luc Rey-Bellet

 Office :  1423 J LGRT
 Phone :  545-6020
 E-Mail :  luc@math.umass.edu
 Office Hours :   M 12:30--13:45, W 11--12:30 or by appointment

Announcement: The class does not meet on Wednesday September 6 and Wednesday September 20. The class will meet on 8:40-9:55 in LGRT 1234 on Friday September 16 and Friday September 30 to make-up these two classes.

Text: The text book for the class is

• Real Analysis. Measure Theory, Integration & Hilbert Spaces, by E. M. Stein & R. Shakarchi. Princeton Lecture Notes in Analysis III, Princeton University Press 2005.
  
  Erratum: Correct proof of Theorem 4.2, Chapter 1 SUP#2 PDF file
  

• Real Analysis: Modern Techniques and their applications, by G.B. Folland. 2nd ed. Wiley 1999.
  
  
• Real Analysis, by H.L. Royden. 3rd ed. Collier Macmillan 1988
  
  
• Measure and Integral: An Introduction to Real Analysis, by R.L. Wheeden and A. Zygmund. M Dekker 1977.
  
  
• A Radical Approach to Lebesgue's Theory of Integration, by D. M. Bressound. (MAA textbooks) Cambridge University Press 2008.
  
  

• Analysis by its History, by E. Hairer and G. Wanner. Undergraduate Texts in Mathematics. Springer
  
  

Syllabus: This is the first part of a 2-semester introduction to real analysis (Math 623-624). The prerequisites for this class is a working knowledge of the basics of analysis taught in a class like M523, see for example Chapter 3 and 4 of Analysis by its History. During the two semesters we will cover

• 1) Measure theory: Lebesgue measure and Integrable functions

• 2) Integration theory: Lebesgue integral, convergence theorems and Fubini theorem

• 3) Differentiation and Integration. Functions of bounded variation.

• 4) Hilbert spaces and Banach spaces . Examples and Applications. Inequalities.

• 5) Abstract measure theory.

Grade: There will be a midterm and a final. Homework will be assigned regularly. Each of the midterms and the combined homeworks will be worth 1/3 of your grade.

Exams:

• Midterm: Friday October 28, 1pm         Room: LGRT 111        
  
  
• Final: Take-home exam, due Friday December 16         Final exam        
  

Homework :

Homework #1 (due on September 26):   Set #1      

Homework #2 (due on October 12 ):   Set #2      

Homework #3 (due on October 26 ):   Set #3      

Homework #4 (due on November 14):   Set #4      

Homework #5 (due on November 30):   Set #5      

Homework #6 (due on December 9):   Set #6