MATHEMATICS (TURKISH, PHD)
PhD	TR-NQF-HE: Level 8	QF-EHEA: Third Cycle	EQF-LLL: Level 8

Course Introduction and Application Information

Course Code	Course Name	Semester	Theoretical	Practical	Credit	ECTS
MAT6011	Functional Analysis II	Fall	3	0	3	9

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:	Departmental Elective
Course Level:
Mode of Delivery:	Face to face
Course Coordinator :	Assoc. Prof. DOĞAN AKCAN
Recommended Optional Program Components:	None
Course Objectives:	This twotier course aims to provide deep understanding of introductory functional analysis.

Learning Outcomes

The students who have succeeded in this course;
The students who succeeded in this course;
o will be able to understand the need for functional analysis and infinite dimensional vector spaces.
o will be able to compare the notions of Metric, Banach and Hilbert spaces.
o will be able to construct formal proofs.
o will be able to compare function,functional and operator.
o will be able to derive the dual space of a given Banach space.
o will be able to explain representation of functionals on Hilbert spaces.

Course Content

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

Weekly Detailed Course Contents

Week	Subject	Related Preparation
1)	Introduction: Metric Space, Open set, Closed set, Neighborhood.
2)	Sequences: Boundedness, Convergence, Cauchy Sequence, Seperability.
3)	Completeness and Completion of Metric Spaces.
4)	Examples. Completeness Proofs.
5)	Vector spaces: Subspace, Dimension, Hamel Basis.
6)	Normed Spaces, Banach Spaces: Normed Space, Banach Space, Further Properties of Normed Spaces.
7)	Finite Dimensional Normed Spaces and Subspaces, Compactness and Finite Dimension.
8)	Linear Operators: Some Properties.
9)	Applications of Bounded and Linear Operators.
10)	Functionals: Linear Functionals, Normed Spaces of Operators
11)	Dual Space: Algebric Dual and Continuous Dual.
12)	Inner Product Spaces, Hilbert Spaces: Inner Product Space Hilbert Space, Further Properties of Inner Product Spaces, Parallelogram Law.
13)	Orthogonal Complements and Direct Sums.
14)	Orthonormal Sets and Sequences, Total Orthonormal Sets and Sequences, representation of Functionals on Hilbert Spaces, Hilbert adjoint Operator.

Sources

Course Notes / Textbooks:	Erwin Kreyszig, “Introductory Functional Analysis with Applications” by Wiley.
References:	Tosun Terzioğlu, Fonksiyonel analizin yöntemleri, İstanbul: Matematik Vakfı 1998

Evaluation System

Semester Requirements	Number of Activities	Level of Contribution
Quizzes	3	% 15
Midterms	2	% 45
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
Study Hours Out of Class	14	3	42
Presentations / Seminar	1	21	21
Quizzes	3	12	36
Midterms	2	20	40
Final	1	19	19
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)	Be able to organize events, for the development of critical and creative thinking and problem solving skills, by using appropriate methods and techniques.
2)	Ability to make individual and team work on issues related to working and social life.
3)	Ability to transfer ideas and suggestions, related to topics about his/her field of interest, written and verball.
4)	Ability to use mathematical knowledge in technology.
5)	To apply mathematical principles to real world problems.
6)	Ability to use the approaches and knowledge of other disciplines in Mathematics.
7)	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.
8)	To be able to link abstract thought that one has to concrete events and to transfer the solutions and examine and interpret the results scientifically by forming experiments and collecting data.