# Functional Analysis (MATH357) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Functional Analysis MATH357 3 0 0 3 6
Pre-requisite Course(s)
N/A
Course Language English N/A Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer, Problem Solving. Assoc. Prof. Dr. Erdal KARAPINAR The aim of the course is providing a familiarity to concepts of the functional analysis, such as norm, compactness and convergence. The students who succeeded in this course; At the end of the course the student is expected to: 1) understand the notions of commutative metric and normed spaces and be able prove the basic facts of these theory, 2) apply this theories to operator theory, 3) know the Hahn-Banach Theorem, 4) know uniform boundedness principle, 5) know open mapping and closed graph theorems. Vector spaces, Hamel basis, linear operators, equations in operators, ordered vector spaces, extension of positive linear functionals, convex functions, Hahn-Banach Theorem, The Minkowski functional, Separation Theorem, metric spaces, continuity and uniform continuity, completeness, Baire Theorem, normed spaces, Banach spaces, the algebra of bounde

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Metric Spaces Open Sets, Closed Sets pp. 2--22
2 Convergence, Cauchy Sequence, Completeness pp. 23--44
3 Vector Spaces, Normed Spaces Banach Spaces pp. 50--66
4 Further Properties of Normed Spaces Finite Dimensional Normed Spaces and Subspaces pp. 67--75
5 Compactness and Finite Dimensional Linear Operators pp. 77--90
6 Bounded and Continuous Linear Operators, Linear Functional pp. 91--110
7 Midterm Exam
8 Linear Operators and Functionals on Finite Dimensional Spaces Normed Spaces of Operators, Dual Spaces pp. 111--125
9 Hahn-Banach Theorem Hahn-Banach Theorem for Complex Valued Vector Spaces and Normed Spaces pp. 213--224
10 Application to Bounded Linear Functionals on C[a,b] pp. 225--230
12 Reflexive Spaces pp. 239-245
13 Midterm Exam
14 Category Theorems Uniform Boundedness Theorem pp. 246--254
15 Strong and Weak Convergence Convergence of Sequence of Operators and Functionals pp.256-268
16 Review

### Sources

Course Book 1. Introductory Functional Analysis with Applications, E. Kreyszig, 1978, John Wiley and Sons Inc. ISBN 0-471-5073-8 2. Elements of the Theory of Functions and Functional Analysis, A.N. Kolmogorov and S.V. Fomin, Dover, NY, 1999. ISBN: 0-486-40683-0 3. Functional Analysis, G.Bachman and L. Narici , Dover, 1991, ISBN: 0-486-40251-7

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments - -
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 60
Final Exam/Final Jury 1 40
Toplam 3 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary X
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. X

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class
Presentation/Seminar Prepration
Project
Report
Homework Assignments
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury
Prepration of Final Exams/Final Jury