MATH 6710 REAL ANALYSIS II
Final exam Week 16. Final exam on Wednesday, April 23, at 12:30-2:30 PM.
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 07 - January 11
Topics for reading
Homework #1 (due to Wednesday , January 16.
- Sections 7.1-7.2 Metric spaces, open and closed sets
- Sections 7.3 -7.4 Continuous Functions. Convergence and completeness.
Week 2: January 14 - January 18
Topics for reading
Homework #2 is due to Wednesday, January 23
- 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 21 - January 25
Topics for reading
Homework #2 is due to Wednesday, January 23. Homework #3 is due to Wednesday, January 30
- Section 7.7 Compactness
- Section 7.8 Baire category
- Section 7.10 Ascoli-Arzel\a theorem
Week 4: January 28 - February 1
Topics for reading
Homework #3 is due to Wednesday, January 30
- Section 7.10 Ascoli-Arzela theorem
- Section 8.1-8.2 Topological spaces: Fundamental Notions. Bases and countability.
Week 5: February 04 - February 8
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 Wednesday, February 13, will include topics from Sections 7.1-7.8, 7.10, 81.-8.3, 10.1-10.2. Homework #5 is due to Friday, February 15.
Week 6: February 11 - February 15
Topics for reading
- Section 10.4 The Closed Graph theorem.
- Section 10.5 Topological vector space
- Section 10.6 Weak topologies
Week 7: February 18 - February 22
Topics for reading
Homework #6 is due to Monday, February 25 .
- Section 10.6 Weak topologies
- Section 10.6 Application of weak topology: existence of minimizers in calculus of variations problems
- Section 10.7 Convexity
Week 8: February 25 - February 29
Topics for reading
- Section 10.6 Application of weak topology: existence of minimizers in calculus of variations problems
- Section 10.7 Convexity (Separation theorem , Th.20)
Homework #6 is due to Monday, February 25 .
Week 9: March 3 - March 7
Spring break - reading week
Week 10: March 10 - March 14
Topics for reading
Homework #7 is due to Friday, March 14.
- Section 10.7 Convexity (Separation theorem , Th.20)
- Section 10.8 Hilbert spaces ( Fourier series, their convergence, examples).
- Part 3 : general measure and integration theory.
- Section 11.1 Measure spaces (probability examples)
Week 11: March 17 - March 21
Topics for reading
Homework #8 is due to Friday, March 21.
- Section 11.1 Measure spaces (probability examples)
- Section 11.2 Measuarable functions, approximation with simple functions
- Section 11.3 Integrals of nonnegative functions
Second mid-term exam on Wednesday, March 19 (Sections 10.2-10.8).
Week 12: March 24 - March 28
Topics for reading
- 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.
Homework #9 is due to Monday, March 31.
Week 13: March 31 - April 04
Topics for reading
- Section 11.5 Signed measures
- Section 11.6 Radon - Nikodym Theorem.
- Section 11.7 L^p spaces.
- Section 12.1 Outer measure and measurability
Week 14: April 7 - April 11
- Section 12.2 Extension of outer measures
- Section 12.4 Product measures, Fubini and Tonelli theorems
- Section 12.9 Hausdorff measures
Homework # 10 is due to Wednesday, April 16.
Week 15: April 14- April 18
Topics for reading
Homework # 10 is due to Wednesday, April 16.
- Section 13.1 Baire and Borel sets
- 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 16 )
Section 7.1 (p.141) # 1 , 2, 3 (bonus problem)
Section 7.2(p.143) # 4, 5, 6, ,7
Homework #2 (due to January 23 )
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 Wednesday, January 30 )
Section 7.7 (p.157) #27, 28,30
Section 7.10 (p.170) # 50, 51 a,b
Homework #4 (due to Wednesday, February 6)
Homework #6 (due to Monday, February 25)
Section 7.8 (p.161) # 31, 32, 37, 38, 43* (bonus problem)
Homework #5 (due to Friday, February 15)
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 14)
Section 10.6 (p. 238) # 38 (a,b,c), 40
Section 10.7 (p.243) 42, 43
Homework #8 (due to March 21)
Section 10.8 (p.249) #50, 51 a,b 53 a,b,c, 54.
Homework # 9 (due to March 28)
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 19)
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 : Wednesday, February 13
- Second mid-term exam: Wednesday, March 19
- Final exam (comprehensive) : Wednesday, April 23 at 12:30 - 2:30.
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 policiesinclude 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 18, 2008
