MATH501 ANALYSIS
| Course Code: | 2360501 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Assoc.Prof.Dr. ÖZCAN YAZICI |
| Offered Semester: | Fall Semesters. |
Course Objectives
At the end of the course, students are expected to learn
- the structure of σ-Algebras
- general measure and integration
- convergence theorems
- product measures and Fubini’s Theorem
- modes of convergence
- decomposition of measures and Radon-Nikodym Theorem
- the statement of Lebesgue Differentiation Theorem
- the basics of the theory of Lp spaces.
Course Content
General measure and integration theory. General convergence theorems. Decomposition of measures. Radon-Nikodym Theorems. Outer measure. Caratheodory extension theorem. Product measures. Fubini's theorem. Riesz representation theorem.
Course Learning Outcomes
After successfully completing this course, students will be able to:
- define and explain key concepts in measure theory, including σ-algebras, measurable functions, and different types of measures.
- define general integral.
- apply major convergence theorems (Monotone Convergence Theorem, Fatou’s Lemma, Dominated Convergence Theorem) to analyze the behavior of sequences of functions.
- use product measures and apply Fubini’s and Tonelli’s Theorems in integration over product spaces.
- apply the Radon-Nikodym Theorem to decompose measures.
- apply the Lebesgue Differentiation Theorem in real analysis contexts.
- work with Lp spaces, including their norms, completeness, duality, and important inequalities (e.g., Hölder’s, Minkowski’s).
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution