Mathematics

Introduction to Functional Analysis(MAT452)

Course Code Course Name Semester Theory Practice Lab Credit ECTS
MAT452 Introduction to Functional Analysis 7 3 0 0 3 5
Prerequisites MAT201, MAT261, MAT262
Admission Requirements MAT201, MAT261, MAT262
Language of Instruction
Course Type Elective
Course Level Bachelor Degree
Course Instructor(s) Ayşegül ULUS aulus@gsu.edu.tr (Email)
Assistant
Objective The primary objective of the course is to teach, without resorting to measure theory, the fundamental tools of functional analysis, namely metric spaces, normed spaces, Banach spaces, inner product spaces, and Hilbert spaces. In addition, we will address contractive mappings in metric spaces as well as applications of approximation theory in normed and Hilbert spaces. Finally, particular attention will be given to the application of these theories to various mathematical, physical, and economic problems, so that students will have studied and explored their concrete applications.
Content Metric Spaces: Review
Normed Spaces, Banach Spaces
Inner Product Spaces, Hilbert Spaces
4 İmportant Theorems about Hilbert Spaces: Projection Theorem, Decomposition Theorem, Riesz Representation Theorem, Hahn-Banach Theorem
Applications: Banach Fixed Point Theorem and Approximation Theorems
Course Learning Outcomes The course learning outcome is to be familiar with basic concepts of functional analysis using the basic tools metric spaces, normed spaces and inner product spaces and to be able to understand the 4 important theorems about Hilbert spaces. They will have become familiar with the applications.
Teaching and Learning Methods Course and Recitation Hours
References Introductory Functional Analysis with Applications, E. Kreyszig, Wiley

An İntroduction to Real Analysis, T. Terzioğlu, ODTÜ

Fonksiyonel Analizin Yöntemleri, T. Terzioğlu, Matematik Vakfı

Fonksiyonel Analiz, E. Şuhubi, İTÜ Vakfı

Bir Analizcinin Defterinden Seçtikleri, T.Terzioğlu, NMK

Real Analysis with Economic Applications, Efe A. Ök, Princeton University Press
Print the course contents
Theory Topics
Week Weekly Contents
1 Metric Spaces: Review
2 Further Examples of Metric Spaces: Sequences Spaces, Function Spaces
3 Completeness
4 Complete Metric Spaces
5 Normed Spaces, Banach Spaces
6 Compactness and Finite Dimension
7 Linear Operators
8 Bounded Operators
9 Linear Functionals
10 Normed Spaces of Operators and Dual Space
11 Inner Product Spaces, Hilbert Spaces
12 4 İmportant Theorems about Hilbert Spaces: Projection Theorem, Decomposition Theorem, Riesz Representation Theorem, Hahn-Banach Theorem
13 Application: Banach Fixed Point Theorem
14 Application: Approximation Theorems
Practice Topics
Week Weekly Contents
Contribution to Overall Grade
  Number Contribution
Contribution of in-term studies to overall grade 2 60
Contribution of final exam to overall grade 1 40
Toplam 3 100
In-Term Studies
  Number Contribution
Assignments 0 0
Presentation 0 0
Midterm Examinations (including preparation) 1 30
Project 1 30
Laboratory 0 0
Other Applications 0 0
Quiz 0 0
Term Paper/ Project 0 0
Portfolio Study 0 0
Reports 0 0
Learning Diary 0 0
Thesis/ Project 0 0
Seminar 0 0
Other 0 0
Make-up 0 0
Toplam 2 60
No Program Learning Outcomes Contribution
1 2 3 4 5
1 understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; X
2 can properly state and use concepts and results of major mathematical interest; X
3 masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; X
4 is able to express one’s mathematical ideas in an organised way both in written and oral forms; X
5 understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); X
6 has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; X
7 has a theoretical and practical knowledge in computer science well adapted for learning a programming language; X
8 has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; X
9 has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; X
10 masters French language as well as other foreign languages, to a level sufficient to study or work abroad. X
Activities Number Period Total Workload
Class Hours 1 3 3
Working Hours out of Class 14 2 28
Assignments 0 0 0
Presentation 1 3 3
Midterm Examinations (including preparation) 1 10 10
Project 1 10 10
Laboratory 0 0 0
Other Applications 0 0 0
Final Examinations (including preparation) 4 10 40
Quiz 0 0 0
Term Paper/ Project 0 0 0
Portfolio Study 0 0 0
Reports 1 10 10
Learning Diary 0 0 0
Thesis/ Project 0 0 0
Seminar 0 0 0
Other 0 0 0
Make-up 0 0 0
Yıl Sonu 1 10 10
Hazırlık Yıl Sonu 0 0 0
Hazırlık Bütünleme 0 0 0
Total Workload 114
Total Workload / 25 4.56
Credits ECTS 5
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