Introduction To Functional Analysis (MAT452) | GSÜ

Content

Objective

The objective of this course is to study the the basic tools for the functional anlysis

Course Content

Banach spaces,
Hilbert spaces,
Hahn Banach theorem
integrability, completeness of Lp spaces
Applications of functional analysis.

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: Complete spaces, compactness
2	Definitions and examples of Banach spaces. Continuous and integrable function spaces
3	Banach spaces, compactness and finite dimension, Ascoli’s theorem
4	Duality in Banach spaces
5	Definitions and examples of Hilbert spaces.
6	Orthogonality and projection. Riesz–Fischer theorem
7	Midterm
8	Fundamental theorem of functional analysis: Zorn's Lemma, Hahn Banach's Theorem
9	Fundamental theorem of functional analysis: Zorn's Lemma, Hahn Banach's Theorem
10	Homework
11	Lp spaces, measurement theory and definition of Lp spaces
12	Lp spaces as Banach spaces, Density in Lp spaces
13	Applications of Functional Analysis: Fourier Transformation and Applications
14	Applications of functional analysis: Sobolev spaces and their properties

Practice Topics
Week	Weekly Contents
1
2
3
4
5
6
7
8
9
10
11
12
13
14

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