Course Code	Semester	Course Name	LE/RC/LA	Course Type	Language of Instruction	ECTS
MB0054		Functional Analysis	2/2/0	DE	Turkish	5

Course Goals	First to teach normed spaces, Banach spaces, inner product spaces, and Hilbert spaces, and then to prove the fundamental theorems of Functional Analysis by giving the concept of a bounded linear operator on this spaces.
Prerequisite(s)	None
Corequisite(s)	None
Special Requisite(s)	None
Instructor(s)	Professor Tunç MISIRLIOĞLU
Course Assistant(s)	Research Assistant Dr. M. Selçuk TÜRER
Schedule	Thursday 13:00-15:00 4B-16, (Recitation) Thursday 15:00-17:00 4C-15/17
Office Hour(s)	Friday 13:00-15:00
Teaching Methods and Techniques	Lecture and Recitation
Principle Sources	B.P. Rynne and M.A. Youngson, Linear Functional Analysis, Second Edition, Springer Undergraduate Mathematics Series, Springer, 2008
Other Sources	-

Course Schedules
Week	Contents	Learning Methods
1. Week	Preliminaries: Linear Algebra; Metric Spaces, Complete Metric Spaces, Baire's Category Theorem, Compact Metric Spaces, Stone-Weierstrass Theorem	Lecture
2. Week	Lebesgue Integration; Minkowskii's and Hölder's Inequalities	Lecture
3. Week	Normed Spaces, Equivalent Norms	Lecture
4. Week	Riesz Lemma, Banach Spaces	Lecture
5. Week	Inner Product Spaces, Cauchy-Schwarz Inequality, Parallelogram Rule, Orthogonality, Gram-Schmidt Algorithm	Lecture
6. Week	Hilbert Spaces, Orthogonal Complements, Convex Set, Orthogonal Decomposition	Lecture
7. Week	Orthonormal Bases in Infinite Dimensional Spaces, Bessel's Inequality, Parseval's Equality	Lecture
8. Week	Continuous and Bounded Linear Operators, Norm of a Bounded Linear Operator, Isometry, Isometric Isomorphism, B(X,Y) Space, Linear Functionals	Lecture
9. Week	Midterm Exam Week	Midterm Exam
10. Week	Inverse of an Operator, Neumann Series, Open Mapping Theorem	Lecture
11. Week	Closed Graph Theorem, Banch's Isomorphism Theorem, Uniform Boundedness Principle	Lecture
12. Week	Dual Spaces, Riesz-Frechet Theorem	Lecture
13. Week	Sublinear Functionals, Seminorms, Hahn-Banach Theorem	Lecture
14. Week	Second Duals, Reflexive Spaces, Dual Operators	Lecture
15. Week	Final Exam Week	Final Exam
16. Week	Final Exam Week	Final Exam
17. Week	Final Exam Week	Final Exam

Assessments
Evaluation tools	Quantity	Weight(%)
Midterm(s)	1	40
Final Exam	1	60

Program Outcomes PO-1 Interpreting advanced theoretical and applied knowledge in Mathematics and Computer Science. PO-2 Critiquing and evaluating data by implementing the acquired knowledge and skills in Mathematics and Computer Science. PO-3 Recognizing, describing, and analyzing problems in Mathematics and Computer Science; producing solution proposals based on research and evidence. PO-4 Understanding the operating logic of computer and recognizing computational-based thinking using mathematics as a discipline. PO-5 Collaborating as a team-member, as well as individually, to produce solutions to problems in Mathematics and Computer Science. PO-6 Communicating in a foreign language, and interpreting oral and written communicational abilities in Turkish. PO-7 Using time effectively in inventing solutions by implementing analytical thinking. PO-8 Understanding professional ethics and responsibilities. PO-9 Having the ability to behave independently, to take initiative, and to be creative. PO-10 Understanding the importance of lifelong learning and developing professional skills continuously. PO-11 Using professional knowledge for the benefit of the society.

Learning Outcomes LO-1 Reminds the needed preliminaries related to the concepts of linear algebra, metric spaces, complete metric spaces, and Riemann Integrals for a course of Functional Analysis. LO-2 Understands the concepts of normed spaces and Banach spaces. LO-3 By understanding inner product spaces, orthogonality, and Hilbert spaces, have a through knowledge of the properties of this spaces. LO-4 Understands continuous and bounded linear operators and their norms. LO-5 Proves the fundamental theorems; which are Open Mapping Theorem, Closed Graph Theorem, and Uniform Boundedness Principle; of Functional Analysis. LO-6 Analyzing Dual Spaces, proves Hahn-Banach Theorem. Recognizes the Second dual dual Reflexive spaces.

Course Assessment Matrix:

	PO 1	PO 2	PO 3	PO 4	PO 5	PO 6	PO 7	PO 8	PO 9	PO 10	PO 11
LO 1
LO 2
LO 3
LO 4
LO 5
LO 6