Advanced Analysis(MATH 501)
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. |
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 |
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 |
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 |
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 |