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Syllabus ( MATH 406 )


   Basic information
Course title: Functional Analysis
Course code: MATH 406
Lecturer: Assoc. Prof. Dr. Ayşe SÖNMEZ
ECTS credits: 7
GTU credits: 0 ()
Year, Semester: 4, Spring
Level of course: First Cycle (Undergraduate)
Type of course: Compulsory
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: Yok
Professional practice: No
Purpose of the course: This course aims to introduce metric spaces, topological spaces, normed and Banach spaces and teach the properties of these spaces, to introduce concepts related to operator theory, and to investigate the existence and uniqueness of operator equations. It also teaches some concepts such as dual spaces, Hahn-Banach extension theorem, weak convergence, inner product spaces and Hilbert spaces.
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Study the properties of metric and topological spaces. Explain and apply basic theorems.

    Contribution to Program Outcomes

    1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
    2. Communicating between mathematics and other disciplines, and building mathematical models for interdisciplinary problems.
    3. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.
    4. Exhibiting professional and ethical responsibility.

    Method of assessment

    1. Written exam
  2. Study a properties of Euclid, Hilbert and Banach spaces. Explain and apply basic theorems

    Contribution to Program Outcomes

    1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
    2. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.

    Method of assessment

    1. Written exam
  3. Define Dual spaces and analyze properties of linear functions and opeators. Explain and apply basic theorems.

    Contribution to Program Outcomes

    1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
    2. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.

    Method of assessment

    1. Written exam
   Contents Up
Week 1: Complete metric spaces, closure.
Week 2: Contraction mapping.
Week 3: Metric and Topological spaces.
Week 4: Compactness. Arzely theorem.
Week 5: Linear normed and topological spaces.
Week 6: Convex functional and Minkowski functional.
Week 7: Euclidean spaces.
Week 8: Midterm exam and solutions.
Week 9: Fourier Series.Riesz-Fischer theorem.
Week 10: Inner product spaces.
Week 11: Dual spaces. The Riesz representation theorem.
Week 12: Strong and Weak Topologies. Strong and Weak Convergence.
Week 13: Bounded and Compact operators.
Week 14: Spectrum and resolvent
Week 15*: -
Week 16*: Final exam.
Textbooks and materials: • A.N.Kolmogorov, S.V. Fomin,Elements of the Theory of Functions and Functional Analysis,Dover Pubns, 1999.
• E. Kreyzig, Introductory Functional Analysis with Applications, John Wiley&Sons, 1978.
• L.A.Lusternik, V.J.Sobolev, “Elements of Functional Analysis”.
• Walter Rudin, “Functional Analysis”.
• John B. Conway “A Course in Functional Analysis”.
• R.E. Edwards, “Functional Analysis : Theory and Applications”.
Recommended readings: • A.N.Kolmogorov, S.V. Fomin,Elements of the Theory of Functions and Functional Analysis,Dover Pubns, 1999.
• E. Kreyzig, Introductory Functional Analysis with Applications, John Wiley&Sons, 1978.
• L.A.Lusternik, V.J.Sobolev, “Elements of Functional Analysis”.
• Walter Rudin, “Functional Analysis”.
• John B. Conway “A Course in Functional Analysis”.
• R.E. Edwards, “Functional Analysis : Theory and Applications”.
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 8 40
Other in-term studies: 0
Project: 0
Homework: 0
Quiz: 0
Final exam: 16 60
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 7 14
Practice, Recitation: 0 0
Homework: 0 0
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 12 1
Mid-term: 3 1
Personal studies for final exam: 18 1
Final exam: 2 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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