MATH 8100
Real Analysis I 
Fall 2006

(Companion Problem Course: MATH 8105)

This website was last updated on January 4, 2007.

Solutions to this year's Real Analysis Qualifying Exam can be found at  http://www.math.uga.edu/~azoff/courses/RealSp07sb.pdf

Final exams have been graded and solutions have been posted at http://www.math.uga.edu/~azoff/courses/fnl8100s.pdf.
The highest score was 182 and the median was 132

Grades have been reported to the registrar and exams are in your mailboxes.

A current list of topics on the Real Analysis Qualifying Exam can be found at http://www.math.uga.edu/~azoff/courses/realqual0510.doc. 
The schedule of MATH 8105 sessions has been updated.

Notes to Problems discussed at previous problem sessions can be found at Aja Johnson's website,  http://www.math.uga.edu/~ajaj/analysis.html

Contents

  • Exams
  • Schedule and assignment summary
  • 8105 sessions
  • Course syllabus
    • Exams

    There were be two hour exams during the term. The first, covering Chapters 1 and 2, was held on September 29.  The median (including corrections) was 74. 

    The second hour exam, covering Chapters 3, 4, 6 from Stein and Lp space material from Royden was turned in on December 4.  The median was  70%.  Solutions are posted at http://www.math.uga.edu/~azoff/courses/8100f06t2s.pdf

    The final exam was held from Noon-3PM on Friday December 8.  The highest score was 182 of 200 and median was 132.  Solutions are posted at
    http://www.math.uga.edu/~azoff/courses/fnl8100s.pdf

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    Assignment Summary

    Number
    		 Due
    		 Chapter
    		 Exercises & Problems 1
    for Class Discussion
    Exercises & Problems1
    to Hand In 
    Bonus
    1
    		 M Aug 28
    		 1 (Stein)
    		 1,3,6,8,9,12,15
    		 2,5,7,11,13,14,16,19
    20,P5 
    2
    		 W Sept 6
    		 1 (Stein)
    		 17,18,27,33,35
    		 21,22,24,25,26,28,32
    		 23,29,P42
    3
    		 M Sept 18
    		 2 (Stein)
    		  2
    		 1,3,6,8,9,10,11,12,P3,A
    		 15
    43
    		 Tu Sept 26
    		 2 (Stein)
    		 16
    		 B,4,7,13,17,18,19,20,21
    		 5,14,P5
    5
    		 Tu Oct 10
    		 6 (Royden4)
    		 3,7,12,13,15,18
    		 1,4,9,10,11,14; test corrections
    		 2,8,16,17
    6 F Oct 20
    		 45 (Stein)
    		 1,11,14,15,19,22
    		 2,3,4,5,6,7,8,10,12
    		 13,18,21,24
    7
    		 F Nov 3
    		 3.1 (Stein)
    		3,4,5,7,8,9
    		 P1,P4
    8
    		 F Nov 10
    		 3.36 (Stein)
    		 6,11,14a,15,18,20,23,25 10,12,13,16,19,22,32
    		 21
    9
    		 M Nov 20
    		 6.1-6.2 (Stein)
    		1,2,3,S1-S67
    		P1
    10
    		6.3-6.4 (Stein)
    		 13
    		 4,5,8,10
    		 9

    1    Unprefixed numbers refer to Exercises in the text.  Numbers prefixed with "P" refer to the Problems which the authors list in separate sections.
    2This is not a trivial problem.  You may look it up in another textbook, but write it up in your own words.
    A. Let (f_n) be a sequence of nonnegative measurable functions on R such that the function sequence (f_n) converges pointwise almost everywhere to an integrable function g.  Suppose also that lim_n \int f_n = \int g, while E is a measurable subset of R.  Prove that lim_n \int_E  f_n = \int_E g .
    B.  Suppose f: is an integrable function on R and the function g defined by g(x)=xf(x) is also integrable.  Define a function F by F(x)= \int f(t) sin(xt) dt.  Carefully use the Dominated Convergence Theorem to show that F is differentiable and express F'(x) as an integral.
    3 Here is a revised pdf file for questions from an earlier problem session.  It includes advice on Assignment 4.
    4  Assignment 5 is based on Chapter 6 of Royden's text; a pdf file is at royden6.pdf
    5  Here is a guide to our coverage of Chapter 4 in Stein.

    6 Here is a guide to over coverage of Chapter 3, Section 3 of Stein.

    7 Supplementary problems for Assignment 9 can be found at http://www.math.uga.edu/~azoff/courses/rprob.pdf

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    8105 Schedule

    A current list of topics on the Real Analysis Qualifying Exam can be found at http://www.math.uga.edu/~azoff/courses/realqual0510.doc.

    Date
    		 Topic
    		 Worksheet
    		 Solutions
    Aug 16
    		 6100-type problems from old Quals
    		 http://www.math.uga.edu/~azoff/courses/8105a.pdf
    Aug 23
    		 Assignment 1 (measure)
    		 http://www.math.uga.edu/%7Eazoff/courses/8100f06.html#assigns
    Aug 30
    		 6100-type problems from old Quals http://www.math.uga.edu/~azoff/courses/8105a.pdf
    Sept 6
    		 Convergence in measure
    		 http://www.math.uga.edu/~azoff/courses/8105b.pdf
    Sept 13
    		 Convergence in measure
    		 http://www.math.uga.edu/~azoff/courses/8105b.pdf
    Sept 20
    		 Limits, DCT, Fubini
    		 http://www.math.uga.edu/~azoff/courses/8105c.pdf
    Sept 27
    		 Review for Test I

    Oct 4
    		 Spring 2004 Qual
    		 http://www.math.uga.edu/graduate/Analysis/Spring2004.pdf http://www.math.uga.edu/%7Eajaj/oct3.pdf
    Oct 11
    		 Fall 2003 Qual
    		 http://www.math.uga.edu/graduate/Analysis/Fall2003.pdf
    		 http://www.math.uga.edu/%7Eajaj/oct11.pdf
    Oct 18
    		 Spring 2005 Qual
    		 http://www.math.uga.edu/graduate/Analysis/Spring2005.pdf
    		 http://www.math.uga.edu/%7Eajaj/oct18.pdf
    Oct 25
    		 Spring 1999 Qual
    		 http://www.math.uga.edu/graduate/Analysis/Spring1999.pdf
    Nov 1
    		 Fall 2004 Qual
    		 http://www.math.uga.edu/graduate/Analysis/Real_Fall2004.pdf
    		 http://www.math.uga.edu/%7Eajaj/nov1.pdf
    Nov 8
    		 Fall 2006 Qual
    		 http://www.math.uga.edu/~azoff/courses/realqual0608.pdf
    		http://www.math.uga.edu/%7Eajaj/nov8.pdf
    Nov 15
    		 Fall 2001 Qual
    		 http://www.math.uga.edu/graduate/Analysis/Fall2001.pdf
    Nov 29
    		 Fall 2000 Qual
    		 http://www.math.uga.edu/graduate/Analysis/Fall2000.pdf

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    MATH 8100/8105 (Azoff) 
      Real Analysis I / Problem Session
    Course Syllabus for Fall 2006

    Web Page  http://www.math.uga.edu/~azoff/courses/8100.html



    Call Numbers 03-136 for MATH 8100; 11-137  for MATH 8105



    Time & Place 1:25 - 2:15  MWF for 8100; 3:30-5:30 W for 8105
    326 Boyd Grad Studies



    Course
    Objectives    		 Understanding of and facility with the Lebesgue integral and its generalizations along with related Banach and Hilbert space topics.  The real analysis qualifying examination covers the material of this course and its prerequisite, MATH 6100.



    Text  Real Analysis: Measure Theory, Integration, & Hilbert Spaces,
    by Elias M. Stein and Rami Shakarchi, Princeton University Press, 2005, ISBN 0-691-11386-6  We will cover the Chapters 1 thru 6, spending about two weeks on each.  This will be supplemented with brief treatments of Lp spaces and Banach spaces, taken from another text; handouts covering this material will be made available.



    Problem Sessions
    		 MATH 8105 meets 3:30-5:30 on Wednesdays.  Sessions will be used to work on class discussion problems, and problems from old qualifying exams.  Students not registered for MATH 8105 are nevertheless welcome to attend.



    Grading Homework 
    Hour Tests (2 @ 100 pts) 
    Comprehensive Final Exam (Noon-3PM on Friday December 8)
    100 points
    200 points
    200 points




    		Homework will be collected about once a week;  no late work will be accepted.   



    Instructor  E. Azoff
       e-mail 
       Phone 
       Office     		 azoff@math.uga.edu
    542-2608 
    443 Boyd 
      Office
      Hours    		 2:30 - 3:30 PM daily, except Monday, October 2

    Assignments for Chapter 1

    Number
    		 Due
    		 Exercises & Problems1 for Class Discussion
    Exercises & Problems1 to Hand In 
    Bonus
    1
    		 F August 25
    		 1,3,6,8,9,12,15
    		 2,5,7,11,13,14,16,19
    20, P5 
    2
    		 F Sept 1
    		 17,18,27,33,35
    		 21,22,24,25,26,28,32,P4
    		 23,29

    1Unprefixed numbers refer to Exercises in the text.  Numbers prefixed with "P" refer to the Problems which the authors list in separate sections.

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