MATH 6710 REAL ANALYSIS II
Schedule for Week 15. Homework # 10 is due to April 23, Friday. Seminar on homework problems on Wednesday, April 21, at 3 p.m., Alavi Commons. Problem-solving session before a final exam on Monday, April 26 at 11 a.m., Alavi Commons.
Course description: This is a second part of two-semester course on real analysis. This semester we study metric, topological and Banach spaces, theory of linear functionals and operators, general measure theory and integration. We consider some applications in calculus of variations and partial differential equations.
A list of problems to work will be placed at the course webpage (see homework problems). Your homework will be collected and graded on regular basis. It is recommended to keep your homework to use in preparation for mid-term and final exams.
Textbook: Real Analysis (Third edition) by H.L.Royden, Prentice Hall.
Office and Classes Hours : see schedule
Week 1: January 11 - January 15
Topics for reading
Homework #1 (due to Thursday , January 21).
- Sections 7.1-7.2 Metric spaces, open and closed sets
- Sections 7.3 -7.4 Continuous Functions. Convergence and completeness.
Week 2: January 18 - January 22
Topics for reading
Homework #1 (due to Thursday , January 21). Homework #2 is due to Thursday, January 28
- Sections 7.4 Convergence and completeness
- Section 7.5 Uniform continuity and uniformity
- Sections 7.6 - 7.7 Subspaces. Compact metric spaces
- Section 7.7 Compactness
Week 3: January 25 - January 29
Topics for reading
Homework #2 is due to Thursday, January 28. Homework #3 is due to Thursday, February 4.
- Section 7.7 Compactness
- Section 7.10 Ascoli-Arzel\a theorem
Week 4: February 1 - February 5
Topics for reading
Homework #3 is due to Thursday, February 4.
- Section 7.10 Ascoli-Arzela theorem
- Section 7.8 Baire category
Week 5: February 8 - February 12
Topics for reading
- Section 8.1-8.2 Topological spaces: Fundamental Notions. Bases and countability.
- Section 8.3 Separation axioms and continuous real-valued functions.
- Section 9.1 Compact topological spaces
- Section 9.3 Products of compact spaces. Tychonoff theorem.
- Section 10.1 Banach spaces
- Section 10.2 Linear operators
- Section 10.3 Linear functionals and Hahn-Banach theorem.
ATTENTION! First mid-term exam on Thursday, February 18, will include topics from Sections 7.1-7.8, 7.10, 81.-8.3, 10.1-10.2. Homework #4 is due to Thursday, February 11.
Week 6: February 15 - February 19
Topics for reading
- Section 10.2 Linear operators
- Section 10.3 Linear functionals and Hahn-Banach theorem
- Section 10.4 The Closed Graph theorem.
- Section 10.5 Topological vector space
- Section 10.6 Weak topologies
Homework #5 is due to Tuesday, February 23
Week 7: February 22 - February 26
Topics for reading
Homework #5 is due to Tuesday, February 23 .
- Section 10.3 Linear functionals and Hahn-Banach theorem
- Section 10.4 The Closed Graph theorem.
- Section 10.5 Topological vector space
Week 8: March 1 - March 5
Spring break - reading week
Week 9: March 8 - March 12
Homework #6 is due to Monday, March 15.Topics for reading
- Section 10.4 The Closed Graph theorem.
- Section 10.5 Topological vector space
- Section 10.6 Weak topologies
.
Week 10: March 15 - March 19
Topics for reading
Homework #7 is due to Friday, March 26.
- Section 10.6 Application of weak topology: existence of minimizers in calculus of variations problems
- Section 10.7 Convexity (Separation theorem , Th.20)
- Section 10.8 Hilbert spaces ( Fourier series, their convergence, examples).
Week 11: March 22 - March 26
Topics for reading
Homework #8 is due to Tuesday, March 30.
- Section 10.8 Hilbert spaces ( Fourier series, their convergence, examples).
- Part 3 : general measure and integration theory
- Section 11.1 Measure spaces (probability examples)
Second mid-term exam on Thursday, April 1 (Sections 10.2-10.8).
Week 12: March 29 - April 2
Topics for reading
Homework # 8 is due to Tuesday, March 30.
Second mid-term exam on Thursday, April 1 (Sections 10.1-10.8).
Week 13: April 5 - April 9
Topics for reading
- Section 11.1 Measure spaces (probability examples)
- Section 11.2 Measuarable functions, approximation with simple functions
- Section 11.3 Integrals of nonnegative functions
- Section 11.3 Integrals of nonnegative functions. Fatou's lemma, Monotone Convergence Theorem, Lebesgue Convergence Theorem
- Section 11.4 General Convergence Theorems.
Week 14: April 12 - April 16
- Section 11.3 Fatou's lemma, Monotone Convergence Theorem, Lebesgue Convergence Theorem
- Section 11.4 General Convergence Theorems.
- Section 11.5 Signed measures
- Section 11.6 Radon - Nikodym Theorem.
- Section 11.7 L^p spaces.
- Section 12.1 Outer measure and measurability
- Section 12.2 Extension of outer measures
- Section 12.4 Product measures, Fubini and Tonelli theorems
Homework # 10 is due to Wednesday, April 21.
Week 15: April 19- April 23
Topics for reading
Homework # 10 is due to Friday, April 23.
- Section 12.4 Product measures, Fubini and Tonelli theorems
- Section 13.2 Regularity of Baire and Borel measures
- Section 13.3 Construction of Borel measures
- Section 13.45 Bounded linear functionals on C(X)
- Review of the course
Homework Problems
Note that problems marked with * are extra.
Homework #1 (due to January 21 )
Section 7.1 (p.141) # 1 , 2, 3 (bonus problem)
Section 7.2(p.143) # 4, 5, 6, ,7
Homework #2 (due to January 28 )
Section 7.3 (p.145) #8, 9 a,b, 10a, 11 a
Section 7.4(p.147) #12, 14, 16, 17a,b
Section Section 7.5 (p.149)20, 24 a
Homework #3 (due to Thursday, February 4 )
Section 7.7 (p.157) #27, 28,30
Section 7.10 (p.170) # 50, 51 a,b
Homework #4 (due to Thursday, February 11)
Homework #6 (due to Friday, March 12)
Section 7.8 (p.161) # 31, 32, 37, 38, 43* (bonus problem)
Homework #5 (due to Thursday, February 25)
Section 8.3 (p.180) # 18 a,b, 23 a
Section 10.1 (p.218) # 1, 3, 5, 6 a,b, 7, 9
Section 10.2 (p.222) 13, 14, 15
Section 10.3 (p.228) #17, 18, 22, 23
Section 10.4 (p.232) #26,27,28
Homework #7 (due to March 26)
Section 10.6 (p. 238) # 38 (a,b,c), 40
Section 10.7 (p.243) 42, 43
Homework #8 (due to March 30)
Section 10.8 (p.249) #50, 51 a,b 53 a,b,c, 54.
Homework # 9 (due to April 15)
Section 11.1 (p. 257) #1, 3 a,b,5 a,b,c
Section 11.2 (p.262) #10,12
Section 11.3 (p.267) # 17, 19, 22 a,b
Homework #10 (due to April 23)
Section 11.7 (p.287) #45, Prove Lemma 28 (p.284)
Section 12.1 (p.291) #1, 2
Section 12.2 (p.299) #7, 11 a
Section 12.4 (p.310) #21, 28, 29, 30
Exams
- First mid-term exam : Thursday, February 18
- Second mid-term exam: Thursday, April 1
- Final exam (comprehensive) : Tuesday, April 27 at 10:15 - 12:15.
Grades and Makeup Policy
Grades will be assigned using a sliding scale. The usual passing level "C" is between 65% and 68%. Someone with 88% may get "A". After each test we'll get an approximate passing level. Do not panic if your grade seems low, come and talk with me about your perfomance.Various numbers below determine the relative weights of exams' and quizzes' grades: Total number of points 500. The final exam - 200 points. Each midterm exam - 80 points, homework grade - 140.In general, there will be NO makeups for exams. If you miss an exam for a valid well-documented reason then arrangements may be made on an individual basis.
Academic Integrity
You are responsible for making yourself aware of and understanding the policies and procedures in the Graduate Catalog (pp. 26-28) that pertain to Academic Integrity. These policies include cheating, fabrication, falsification and forgery, multiple submission, plagiarism, complicity and computer misuse. If there is reason to believe you have been involved in academic dishonesty, you will be referred to the Office of Student Conduct. You will be given the opportunity to review the charge(s). If you believe you are not responsible, you will have the opportunity for a hearing. You should consult with me if you are uncertain about an issue of academic honesty prior to the submission of an assignment or test.
Comments, questions, problems to ledyaev@wmich.edu .Last modified : April 16, 2010
