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;
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| 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 | Alanında, bağımsız olarak, bir problem kurgulayabilir, çözüm yöntemi geliştirerek problemi çözebilir ve sonuçları değerlendirebilir | X | ||||
| 2 | Matematiğin temel alanlarında ve kendi uzmanlığı olarak seçtiği alanda gerekli alt yapıyı oluşturur. | X | ||||
| 3 | Matematik literatürünü ve özel olarak kendi araştırma konusu ile ilgili ulusal ve uluslararası güncel yayınları takip edebilir ve bunlardan kendi araştırma konusu ile ilgili olanları çalışmalarında kullanabilir | X | ||||
| 4 | Bilimsel etik değerleri ve kuralları dikkate alır ve mesleki ve toplumsal yaşamda kullanabilir | X | ||||
| 5 | Kendi çalışmalarının sonuçlarını veya belli bir konudaki güncel çalışmaları ve bulguları, çeşitli bilimsel toplantılarda topluluk önünde Türkçe ve İngilizce olarak sunabilir ve tartışmalara katılabilir. | X | ||||
| 6 | Gerek bireysel, gerek bir çalışma grubunun üyesi olarak çalışabilme becerisini geliştirir | X | ||||
| 7 | Yaratıcı ve eleştirel düşünme, problem çözme, özgün bir çalışma üretme becerisini geliştirir. Bilimsel gelişmeleri takip eder, özümsediği bilgilerin analiz, sentez ve değerlendirmesini yapabilir. | X | ||||
| 8 | Kazandığı bilgi, beceri ve yetkinlikleri yaşam boyu geliştirmeye açık olur. | X | ||||
| 9 | Alanında özümsediği bilgiyi ve problem çözme yeteneğini disiplinler arası çalışmalarda uygulayabilir; karşılaşılan problemleri matematiksel modellerle ifade ederek, matematiksel bakış açısı ile farklı çözüm yöntemleri önerir. | X | ||||
| 10 | Matematik temelli yazılımları, bilişim ve iletişim teknolojilerini bilimsel amaçlı kullanabilir. | 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 | ||