Master Program in Mathematics

Advanced Analysis(MATH 501)

Content
Course Code Course Name Semester Theory Practice Lab Credit ECTS
MATH 501 Advanced Analysis 1 3 0 0 3 8
Prerequisites
Admission Requirements
Language of Instruction Turkish
Course Type Compulsory
Course Level Masters Degree
Course Instructor(s) SUSUMU TANABE tanabesusumu@hotmail.com (Email) Gönenç ONAY gonay@gsu.edu.tr (Email)
Assistant
Objective The course aims to cover some parts of the content of Mat 101,102, 201, 202, 301,331 and 452 given in the undergraduate level at Galatasaray University. We try to understand definitions, theorems, and proofs of some results in Real Analysis. We don't prove everything but will try to get a deeper understanding and hope to consolidate your understanding in Real Analysis.
Content 1. Analytic functions, harmonic functions.
2. Cauchy-Riemann equation.
3. Cauchy integral theorem
4. Cauchy integral formula
5. Riemann Integral for several variable functions, Fubini's theorem.
6. Lebesgue Outer measure. Measurable sets in R, then in R^n
7. Measurable Functions
8. Completion of a Measure space
9. Lebesgue Integral
10. Properties of Lebesgue Integral
11. Comparison of Riemann and Lebesgue Integrals, Convergence Theorems
12. Lebesgue Integral in R^n, Fubinis'theorem for Lebesgue Integral
13. L^p spaces, Convolution
14. Jordan and Hahn Decompositions, Radon–Nikodym Theorem
Course Learning Outcomes ÖÇ 1: Teaching the main concepts of Real analysis and having ability to apply to other branches of analysis.
Teaching and Learning Methods
References 1) A. W. Knapp, Basic Real Analysis, with an appendix ”Ele- mentary Complex Analysis”, Digital Second Edition, 2016.
2) G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 1999.
3) W. Rudin, Real and Complex Analysis, McGraw-Hill Inc., 1966.
Print the course contents
Weekly Contents
Theory Topics
Week Weekly Contents
1 Finite and infinite sets, countability.
2 Interchange of Limits, Pointwise Convergence, Uniform Convergence
3 Riemann Integral
4 Metric Spaces, Open/Closed sets, Compactness, Completeness, Examples: C(S) and B(S)
5 Riemann Integral for several variable functions, Fubini's theorem.
6 Lebesgue Outer measure. Mesaurable sets in R, then in R^n
7 Measurable Functions
8 Completion of a Measure space
9 Lebesgue Integral
10 Properties of Lebesgue Integral / Midterm
11 Comparison of Riemann and Lebesgue Integrals, Lebesgue Convergence Theorems
12 Lebesgue Integral in R^n, Fubinis'theorem for Lebesgue Integral
13 L^p spaces, Convolution
14 Jordan and Hahn Decompositions, Radon–Nikodym Theorem
Practice Topics
Week Weekly Contents
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Assessment System
Contribution to Overall Grade
  Number Contribution
Contribution of in-term studies to overall grade 2 50
Contribution of final exam to overall grade 1 50
Toplam 3 100
In-Term Studies
  Number Contribution
Assignments 2 15
Presentation 1 5
Midterm Examinations (including preparation) 1 30
Project 0 0
Laboratory 0 0
Other Applications 0 0
Quiz 0 0
Term Paper/ Project 0 0
Portfolio Study 0 0
Reports 0 0
Learning Diary 0 0
Thesis/ Project 0 0
Seminar 0 0
Other 0 0
Toplam 4 50
Relation of Proficiency
No Program Learning Outcomes Contribution
1 2 3 4 5
1 X
2 X
3 X
4 X
5 X
6 X
7 X
8 X
9 X
10 X
11 X
12 X
13 X
14 X
15 X
16 X
17 X
ECTS
Activities Number Period Total Workload
Class Hours 14 3 42
Working Hours out of Class 15 5 75
Final Examinations (including preparation) 1 71 71
Total Workload 188
Total Workload / 25 7,52
Credits ECTS 8
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