1	Course Title:	REAL ANALYSIS II
2	Course Code:	MAT5102
3	Type of Course:	Optional
4	Level of Course:	Second Cycle
5	Year of Study:	1
6	Semester:	2
7	ECTS Credits Allocated:	6
8	Theoretical (hour/week):	3
9	Practice (hour/week) :	0
10	Laboratory (hour/week) :	0
11	Prerequisites:	none
12	Recommended optional programme components:	None
13	Language:	Turkish
14	Mode of Delivery:	Face to face
15	Course Coordinator:	Prof. Dr. OSMAN BİZİM
16	Course Lecturers:	Prof. Dr. Osman Bizim
17	Contactinformation of the Course Coordinator:	Uludağ Üniversitesi, Fen-Edebiyat Fakültesi Matematik Bölümü, Görükle Bursa-TÜRKİYE 0 224 294 17 57 / obizim@uludag.edu.tr
18	Website:
19	Objective of the Course:	The aim of this course is to review student’s undergradute analysis courses and to correct the deficiencies. So students can be su have succsessful in graduate studies.
20	Contribution of the Course to Professional Development

21

Learning Outcomes:

1 Learns measure and its applications.; 2 Learns measurable sets, measurable functions and the Lebesgue measure.; 3 Learns the Lebesgue integral and its properties.; 4 Learns Lp-spaces and convex functions.; 5 Learns Hilbert spaces, inner-product spaces and linear functionals.; 6 Learns orthonormal sets and trigonometric series.; 7 Learns the Fourier series of continous functions.;

22	Course Content:

Week	Theoretical	Practical
1	Set functions and their properties
2	Measura function, measure space and their properties
3	Construction the Lebesgue and the Borel measure and their properties
4	Measurable functions
5	Simple functions and their properties
6	The Lebesgue integral of simple functions and their properties
7	The Lebesgue covergence theorem and its applications
8	The integral of complex functions and their properties
9	The Riesz-Fischer theorem and its applications
10	Lp-spaces and their properties
11	Convex functions and their properties
12	Hilbert spaces inner-product spaces and linear functionals.
13	Orthonormal sets and trigonometric series and their properties
14	Banach spaces and the Fourier series of continous functions.

23	Textbooks, References and/or Other Materials:	[1] Principles of Mathematical Analysis, W. Rudin, [2] Real and Complex Analysis, W. Rudin, [3] Real Analysis, H. L. Royden, [4] Introduction to Real Analysis, W. F. Trench.
24	Assesment

TERM LEARNING ACTIVITIES	NUMBER	PERCENT
Midterm Exam	0	0
Quiz	0	0
Homeworks, Performances	0	0
Final Exam	1	100
Total	1	100
Contribution of Term (Year) Learning Activities to Success Grade		0
Contribution of Final Exam to Success Grade		100
Total		100
Measurement and Evaluation Techniques Used in the Course
Information

25

ECTS / WORK LOAD TABLE

Activites	NUMBER	TIME [Hour]	Total WorkLoad [Hour]
Theoretical	14	3	42
Practicals/Labs	0	0	0
Self Study and Preparation	14	5	70
Homeworks, Performances	0	0	0
Projects	0	0	0
Field Studies	0	0	0
Midtermexams	0	0	0
Others	14	5	70
Final Exams	1	43	43
Total WorkLoad			225
Total workload/ 30 hr			7,5
ECTS Credit of the Course			7,5

26	CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS
PQ1 PQ2 PQ3 PQ4 PQ5 PQ6 PQ7 PQ8 PQ9 PQ10 LO1 5 5 5 5 5 5 5 5 5 5 LO2 5 5 5 5 5 5 5 5 5 5 LO3 5 5 5 5 5 5 5 5 5 5 LO4 5 5 5 5 5 5 5 5 5 5 LO5 5 5 5 5 5 5 5 5 5 5 LO6 5 5 5 5 5 5 5 5 5 5 LO7 5 5 5 5 5 5 5 5 5 5

LO: Learning Objectives

PQ: Program Qualifications

Contribution Level:

1 Very Low

2 Low

3 Medium

4 High

5 Very High