Topological Vector Spaces
Mathematics
Institute of Natural and Applied Sciences
Third Cycle (Doctorate Degree)
| Course Unit Title | Course Unit Code | Type of Course Unit | Level of Course Unit | Year of Study | Semester | ECTS Credits |
|---|---|---|---|---|---|---|
| Topological Vector Spaces | MAT621 | Elective | Doctorate degree | 1 | Fall | 8 |
Name of Lecturer(s)
Prof. Dr. Halis AYGÜN
Prof. Dr. Çiğdem GÜNDÜZ
Associate Prof. Dr. Vildan ÇETKİN
Associate Prof. Dr. Banu PAZAR VAROL
Learning Outcomes of the Course Unit
1) Identify the notion of a topological vector space
2) Explain the completeness property of a topological vector space
3) Explain the bounded sets, balanced sets and learn the lconcepts of local convex and normed spaces of topological vector spaces
4) Explain the fundamental theorems and corollaries such as Baire Category Theorem, Banach-Steinhaus Theorem, Open Mapping and Closed Function Theorems, Hahn-Banach Theorem
5) State the concepts of subspace, product and quotient space of topological vector spaces and some of their properties
Program Competencies-Learning Outcomes Relation
| Program Competencies | ||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| Learning Outcomes | ||||||||
| 1 | No relation | No relation | No relation | No relation | No relation | No relation | No relation | |
| 2 | No relation | No relation | No relation | No relation | No relation | No relation | No relation | |
| 3 | No relation | No relation | No relation | No relation | No relation | No relation | No relation | |
| 4 | No relation | No relation | No relation | No relation | No relation | No relation | No relation | |
| 5 | No relation | No relation | No relation | No relation | No relation | No relation | No relation | |
Mode of Delivery
Face to Face
Prerequisites and Co-Requisites
None
Recommended Optional Programme Components
Linear Algebra 1, Linear Algebra 2, Topology 1, Topology 2
Course Contents
Candidates are provided with in-depth knowledge on linear spaces, linear mappings, separability, metrization, semi-norms, fundamental concepts and properties in topological vector spaces, completeness property, subspaces, product and quotient spaces, bounded sets, local convex spaces, normed spaces, boundedness and continuity, Baire category theorem, Banach-Steinhaus theorem, open mapping and closed graph theorem, Hahn-Banach theorem, weak topologies, dual spaces of normed spaces, dual and compact operators.
Weekly Schedule
1) Linear spaces.2) Linear (vector) spaces and linear operators.
3) Separable property.
4) Separability and metrizability.
5) Metrizability and semi norms.
6) Fundamental properties and notions of topological vector spaces.
7) Fundamental properties and concepts in topological vector spaces and completeness property.
8) Midterm exam.
9) Completeness property and subspaces.
10) Product and quotient spaces.
11) Bounded and total bounded sets, local convex spaces.
12) Normed spaces, boundedness and continuity, Baire category theorem and Banach-Steinhaus theorem.
13) Open mapping theorem and closed graph theorem.
14) Hahn-Banach theorem and weak topologies.
15) Dual spaces of normed spaces, dual and compact operators.
16) Final exam
Recommended or Required Reading
Planned Learning Activities and Teaching Methods
Assessment Methods and Criteria
Language of Instruction
Turkish
Work Placement(s)
Not Required