MAT6011 Functional Analysis IIBahçeşehir UniversityDegree Programs MATHEMATICS (TURKISH, PHD)General Information For StudentsDiploma SupplementErasmus Policy StatementNational Qualifications
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 8
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. ERSİN ÖZUĞURLU
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.


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
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