ECTS - Measure Theory and Functional Analysis
Measure Theory and Functional Analysis (MATH651) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Measure Theory and Functional Analysis | MATH651 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Ph.D. |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer. |
| Course Lecturer(s) |
|
| Course Objectives | This is a beginning graduate course that is a blend of Measure Theory and selected topics in Functional Analysis. The aim is to provide a firm background and introduce modern ideas and techniques to the serious student who wants to pursue a research career in Functional Analysis, while students who want to build more on undergraduate Functional Analysis are welcome as well. We will cover the material as rigorously as possible, and therefore, for the convenience of the students, we do a rather large review of topological spaces, measure spaces and integration, and also Banach and Hilbert spaces and their operators. We will do an analysis of C(X), where X is a compact Hausdorff space, and finally we will cover the Continuous and Borel Functional Calculus and the Spectral Theorem for bounded self adjoint operators of a separable Hilbert Space. |
| Course Learning Outcomes |
The students who succeeded in this course;
|
| Course Content | Topological and metric spaces, measure spaces and integration, Banach spaces and the four structural Banach space theorems, an analysis of C(X) and the Stone - Weierstrass theorem and its applications, Hilbert spaces and Hilbert bundles, the continuous and Borel functional calculus and the spectral theorem. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | Review of Topological and Metric Spaces | [1], 1.2-1.3, 1.6-1.7 [2], 4.1-4.2 |
| 2 | Review of Measure spaces and the Lebesgue Measure | [1], 2.1-2.2, 2.4 |
| 3 | Review of Measure spaces and the Lebesgue Measure | [1], 2.1-2.2, 2.4 |
| 4 | Review of the Lebesgue Integral and Convergence Theorems | [1], 2.5-2.6 |
| 5 | Review of the Lebesgue Integral and Convergence Theorems | [1], 2.5-2.6 |
| 6 | Review of normed and Banach Spaces and bounded linear operators | [1], 3.1 [2], 5.1 |
| 7 | Review of 4 fundamental theorems of Banach Spaces: Hahn - Banach Theorem, Open Mapping and Closed Graph Theorems and Principle of Uniform Boundedness | [1], 3.3-3.4 [2], 5.2-5.3 |
| 8 | C(X) and subalgebras: the Stone - Weierstrass Theorem | [1], 3.5 |
| 9 | Ideals and homomorphisms of C(X) | [1], 3.7 |
| 10 | L^p spaces | [2], 6.1 |
| 11 | Review of Hilbert Spaces | [1], 5.1 [2], 5.5 |
| 12 | Hilbert bundles and spectral measures | [1], 5.2 |
| 13 | Hilbert bundles and spectral measures | [1], 5.2 |
| 14 | Continuous functional calculus | [1],5.3- 5.4 |
| 15 | Continuous functional calculus | [1],5.3- 5.4 |
| 16 | Borel functional calculus and the Spectral theorem | [1], 5.5 |
Sources
| Course Book | 1. N. Weaver, Measure Theory and Functional Analysis, World Scientific, 2013. |
|---|---|
| 2. G. B. Folland, Real Analysis Modern Techniques and Their Applications, Wiley, 1999. |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 4 | 40 |
| Presentation | 1 | 10 |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 20 |
| Final Exam/Final Jury | 1 | 30 |
| Toplam | 7 | 100 |
| Percentage of Semester Work | 70 |
|---|---|
| Percentage of Final Work | 30 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Is independently able to build a problem in the area of study, solve the problem by developing solution techniques and assess the solutions. | X | ||||
| 2 | Is capable of creating a groundwork in the fundamental branches of mathematics as well as in his/her research area | X | ||||
| 3 | follows the latest national and international literature in Mathematics and in his/her area of research; and uses them in his/her related studies | X | ||||
| 4 | observes and adopts the scientific ethical values in his/her professional and social life | X | ||||
| 5 | presents in Turkish and English in academic/scientific events the results of his/her research or the latest studies and findings on a special topic and participates in discussions | X | ||||
| 6 | Develops skills to work independently or as a member of a team | X | ||||
| 7 | Develops competences in the areas of creative and critical thinking, problem solving and producing original studies. Follows recent scientific studies, is capable of making an analysis, synthesis and assessment of the knowledge acquired | X | ||||
| 8 | Is open to lifelong improvement of his/her acquired knowledge, skills and competences. | X | ||||
| 9 | Is able to apply the acquired knowledge and problem-solving skills to interdisciplinary studies, proposes different solution methods to problems in terms of mathematical models and from a mathematical point of view | X | ||||
| 10 | Uses the mathematical based softwares, informatics and communication technologies for scientific purposes | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | 16 | 3 | 48 |
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 2 | 28 |
| Presentation/Seminar Prepration | 1 | 5 | 5 |
| Project | |||
| Report | |||
| Homework Assignments | 4 | 6 | 24 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 10 | 10 |
| Prepration of Final Exams/Final Jury | 1 | 10 | 10 |
| Total Workload | 125 | ||