(MATH 501) | GSÜ - ECTS Information Guide

Content

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.

Course 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.

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.

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
Activities	Number	Contribution
Contribution of in-term studies to overall grade	2	50
Contribution of final exam to overall grade	1	50
Total	3	100

In-Term Studies
Activities	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
Total	4	50

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	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