Introduction To Functional Analysis (MAT452) | GSÜ

Content

Objective

The objective of this course is to study the the basic tools for the functional anlysis without any refernce to measure theory

Course 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

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

Teaching and Learning Methods

Course and Recitation Hours

References

Introductory Functional Analysis and Applications, Erwin Kreyszig

Weekly 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	Orthoganal Complements and Orhonormal Sets and Sequences
13	4 İmportant Theorems about Hilbert Spaces: Projection Theorem, Decomposition Theorem, Riesz Representation Theorem, Hahn-Banach Theorem
14	4 İmportant Theorems about Hilbert Spaces: Projection Theorem, Decomposition Theorem, Riesz Representation Theorem, Hahn-Banach Theorem

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	1	60
Contribution of final exam to overall grade	1	40
Total	2	100

In-Term Studies
Activities	Number	Contribution
Assignments	0	0
Presentation	0	0
Midterm Examinations (including preparation)	1	30
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	2	30
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
Total	3	60

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	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

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	4	56
Working Hours out of Class	14	2	28
Midterm Examinations (including preparation)	1	10	10
Final Examinations (including preparation)	1	20	20
Quiz	2	6	12
Total Workload			126
Total Workload / 25			5.04
Credits ECTS			5