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| MATL7195 | Basic of Real Analysis | 3+0+0 | ECTS:7.5 | | Year / Semester | Fall Semester | | Level of Course | Third Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Prof. Dr. Zameddin İSMAİLOV | | Co-Lecturer | Prof. Dr. Bahadır.Ö.Güler | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | Explanation of Lebesgue Measure and Integral Theory |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | learn the Lebesge measure and integral theory | 2,3 | 1 | | PO - 2 : | see the relations betwen Lebesgue integral and Riemann and Riemann Stieltjes integrals | 2,3 | 1 | | PO - 3 : | use to the theory of differential equetions and probability theory | 2,3 | 1 | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), PO : Learning Outcome | | |
| Introduction, Fundamental concepts of Measurable Functions and Measures, Lebesgue Integration and Basic Results, Lebesgue Spaces, Holder and Minkowski Inequalities, Completness of Lebegue Spaces, Modes of Convergences, Convergence Theorems, Hahn Decomposition Theorem, Radon-Nikodym Theorem, Riesz Repsentation Theorem |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Introduction | | | Week 2 | Measure and Meansurable Functions, Basic Results | | | Week 3 | Modes Convergences, Convergences in Meusure | | | Week 4 | Lebesgue Measure | | | Week 5 | Monotone Convergence and Lebesgue Dominited Convergence Theorems | | | Week 6 | Lebesgue Integration whith parameter | | | Week 7 | Lebesgue Integration and Fubini's Theorem | | | Week 8 | Mid-term exam | | | Week 9 | Banach and Hilbert Spaces | | | Week 10 | Lebesgue spaces and Completeness | | | Week 11 | Convergence in Lebesgue spaces | | | Week 12 | Hahn Decomposition Theorem | | | Week 13 | Radon-Nikodym's Theorem | | | Week 14 | Riesz Repsentation Theorem | | | Week 15 | Applications | | | Week 16 | End-of-term exam | | | |
| 1 | Aliprantis,C.D.,Burkinshaw,O.1990;Principles of Real Analysis,Academic Press,San Diego | | | 2 | Bartle, R.G. 1966; The Element of Integration, John Wiley Sons, New York | | | |
| 1 | Halmos, P.R. 1950; Measure Theory, D. Van Nostrand Comp., New York | | | 2 | Balcı, M., 1998; Ankara Üniversitesi Fen Fakültesi Matematik Bölümü Yatınları, Ankara | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 18/04/2022 | 2 | 30 | | Quiz | 13 | 09/05/2022 | 1.5 | 20 | | End-of-term exam | 15 | 06/06/2022 | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 4 | 14 | 56 | | Sınıf dışı çalışma | 8 | 14 | 112 | | Arasınav için hazırlık | 4.5 | 2 | 9 | | Arasınav | 2 | 1 | 2 | | Kısa sınav | 1.5 | 1 | 1.5 | | Dönem sonu sınavı için hazırlık | 4.5 | 3 | 13.5 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 196 |
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