APPLIED MATHEMATICS (TURKISH, NON-THESIS)
Master TR-NQF-HE: Level 7 QF-EHEA: Second Cycle EQF-LLL: Level 7

Course Introduction and Application Information

Course Code Course Name Semester Theoretical Practical Credit ECTS
MAT5009 Functional Analysis I Spring 3 0 3 12
This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester.

Basic information

Language of instruction: Turkish
Type of course: Must Course
Course Level:
Mode of Delivery: Face to face
Course Coordinator : Assoc. Prof. ERSİN ÖZUĞURLU
Recommended Optional Program Components: None
Course Objectives: This course provides deep understanding of introductory functional analysis. The objective of this course is to cover fundamental theorems of functional analysis such as HahnBanach theorem, Open mapping theorem, Closed graph theorem, Baire’s category theorem, Banach fixed point theorem, and their applications.

Learning Outcomes

The students who have succeeded in this course;
The students who succeeded in this course;
o will be able to know the essential theorems such as Open mapping theorem, Closed graph theorem, Baire’s category theorem, and Banach fixed point theorem of functional analysis and to be aware of the important applications of these theorems.
o will be able to compare strong and weak convergence.
o will be able to know the differences between convergences of sequences of operators and functionals.

Course Content

This course aims to teach basic theory and applications of Functional Analysis.

Weekly Detailed Course Contents

Week Subject Related Preparation
1) Hilbert Spaces: HilbertAdjoint Operator.
2) SelfAdjoint, Unitary and Normal Operators.
3) Fundamental Theorems for Normed and Banach Spaces: Zorn’s Lemma,HahnBanach Theorem.
4) HahnBanach Theorem for Complex and Normed Spaces, and It’s Application to C[a,b].
5) Adjoint Operator.
6) Reflexive Spaces.
7) Category Theorem, Uniform Boundedness Theorem, and Applications.
8) Convergence: Strong and Weak Convergence.
9) Convergence of Sequences of Operators and Functionals.
10) Open Mapping theorem.
11) Closed Linear Operators and Closed Graph Theorem.
12) Banach Fixed Point Theorem: Application of Banach’s Theorem to Linear Equations.
13) Application of Banach’s Theorem to Linear Equations, Differential Equations, and Integral Equations.
14) Approximation Theory: Approximation in Normed Spaces, Uniqueness and Strict Convexity. Uniform Approximation, Chebyshev Polynomials, Approximation in Hilbert Spaces.

Sources

Course Notes / Textbooks: Walter Rudin, Functional Analysis 2/E, International Series in Pure and Applied Mathematics.
References: Erwin Kreyszig, “Introductory Functional Analysis with Applications” by Wiley.

Evaluation System

Semester Requirements Number of Activities Level of Contribution
Homework Assignments 7 % 30
Presentation 1 % 30
Final 1 % 40
Total % 100
PERCENTAGE OF SEMESTER WORK % 60
PERCENTAGE OF FINAL WORK % 40
Total % 100

ECTS / Workload Table

Activities Number of Activities Duration (Hours) Workload
Course Hours 14 3 42
Presentations / Seminar 1 40 40
Homework Assignments 7 13 91
Final 1 27 27
Total Workload 200

Contribution of Learning Outcomes to Programme Outcomes

No Effect 1 Lowest 2 Low 3 Average 4 High 5 Highest
           
Program Outcomes Level of Contribution
1) Ability to assimilate mathematic related concepts and associate these concepts with each other.
2) Ability to gain qualifications based on basic mathematical skills, problem solving, reasoning, association and generalization.
3) Be able to organize events, for the development of critical and creative thinking and problem solving skills, by using appropriate methods and techniques.
4) Ability to make individual and team work on issues related to working and social life.
5) Ability to transfer ideas and suggestions, related to topics about his/her field of interest, written and verball.
6) Ability to use mathematical knowledge in technology.
7) To apply mathematical principles to real world problems.
8) Ability to use the approaches and knowledge of other disciplines in Mathematics.
9) Be able to set up and develope a solution method for a problem in mathematics independently, be able to solve and evaluate the results and to apply them if necessary.
10) To apply mathematical principles to real world problems.
11) To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively.
12) To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself.