The aim of studying and bibliography

The main aim of study: The aim of education is first of all delivery of the honest knowledge from functional analysis. During the classes of this subject, students are instructed to basic structures and methods of proving of theorems, which are applied in this area. Moreover, students to get to know fundamental tools of functional analysis, that are used in modern mathematics.

The general information about the module: Topics discussed in the module: Algebraic sum, a basis of a vector space, a quotient space, a convex set, a balanced set. Extremal points. Normed spaces. Examples of sequence and function Banach spaces. Holder and Minkowski inequalities. Separability. Isomorphism of spaces and equivalence of two norms. Schauder basis. Completeness of normed spaces. Baire theorem. Product and quotient spaces. Criterions of compactness in some Banach spaces. Unitary spaces. Schwarz inequality. Hilbert spaces. Theorems on the best approximation. Gramm determinant. Orthogonality. Schmidt orthogonalization theorem. Orthogonal complements and projections. Orthonormal bases. Fourier coefficients and orthonormal series. Linear operators, bounded linear operators, the norm of operator, the space of linear bounded operators. Some classes of operators: isometry, isomorphism, finite dimensional operators, adjoint operators, compact operators. Banach-Steinhaus theorem, open mapping theorem, inverse operator theorem. Closed graph theorem.

Bibliography required to complete the module

Bibliography used during lectures

1	Julian Musielak	Wstęp do analizy funkcjonalnej	Warszawa PWN.	1989
2	Walter Rudin	Analiza funkcjonalna	Wydawnictwo Naukowe PWN, Warszawa.	1998

Bibliography used during classes/laboratories/others

1

Stanisław Prus, Adam Stachura

Analiza funkcjonalna w zadaniach

Warszawa Wyd. nauk. PWN.

2009

Bibliography to self-study

1

Julian Musielak

Wstęp analizy funkcjonalnej

Warszawa PWN.

1989

Basic requirements in category knowledge/skills/social competences

Formal requirements: The student satisfies the formal requirements set out in the study regulations

Basic requirements in category knowledge: basic knowledge of linear algebra, topology of metric spaces, differential and integral calculus of a function of single variable, measure theory

Basic requirements in category skills: ability of making algebraic operations, calculating limits, investigating monotonicity of a function, ability of operating with basic topological notion

Basic requirements in category social competences: ability of individual and group learning, awareness of the level of own knowledge and awareness of necessity of self-education

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with PRK
01	student can examine convergence of a sequence of vectors in a normed space	lectures, problem exercises	written test, exam	K_W01+ K_W02+ K_U01+ K_U05+++ K_U09+++ K_U14++	P7S_UW P7S_WG P7S_WK
02	student can investigate basic topological properties (compactness, closedness, openness) topological properties of some subsets of Banach spaces	lectures, problem exercises	written test, exam	K_W01+ K_W03+ K_W04++ K_W05+ K_W07+ K_U03+ K_U07+ K_U08+++	P7S_UW P7S_WG
03	student knows how to apply appropriate methods of the best approximation to the simple problems of minimization	lectures, problem exercises	written test, exam	K_U02+ K_U04+ K_U07+ K_U10+ K_U13++ K_U15+ K_U16+	P7S_UK P7S_UO P7S_UU P7S_UW
04	student can calculate the norm of a linear bounded operator and he knows how to examine some of its properties	lectures, problem exercises,	written test, exam	K_U04+ K_U17+ K_K01+ K_K02+ K_K04+ K_K07+	P7S_KK P7S_KO P7S_KR P7S_UW

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
1	TK01	1. Linear spaces - reminder of notions. Algebraic sum, a basis of a vector space, a quotient space, a convex set, a balanced set. Extremal points. 2. Banach spaces. Normed spaces. Examples of sequence and function Banach spaces. Holder nad Minkowski inequalities.	W01-W06, C01-C09	MEK01 MEK02
1	TK02	3. Topological properties of normed spaces. Separability. Isomorphism of spaces and equivalence of two norms. Schauder basis. Completeness of normed spaces. Baire theorem. Product and quotient spaces. Criterions of compactness in some Banach spaces.	W07-W014, C10-C021	MEK01 MEK02
1	TK03	4. Hilbert spaces. Unitary spaces. Schwarz inequality. Hilbert spaces. Theorems on the best approximation. Gramm determinant. Orthogonality. Schmidt orthogonalization theorem. Orthogonal complements and projections. Orthonormal bases. Fourier coefficients and orthonormal series.	W15-W20, C22-C33	MEK03
1	TK04	5. Linear operators. Linear operators, bounded linear operators, the norm of operator, the space of linear bounded operators. Some classes of operators: isometry, isomorphism, finite dimensional operators, adjoint operators, compact operators. Banach-Steinhaus theorem, open mapping theorem, inverse operator theorem. Closed graph theorem.	W21-W30, C34-C45	MEK01 MEK02 MEK04

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 1)		contact hours: 30.00 hours/sem.	complementing/reading through notes: 2.00 hours/sem. Studying the recommended bibliography: 10.00 hours/sem.
Class (sem. 1)	The preparation for a Class: 10.00 hours/sem. The preparation for a test: 20.00 hours/sem.	contact hours: 45.00 hours/sem.	Finishing/Studying tasks: 5.00 hours/sem.
Advice (sem. 1)		The participation in Advice: 3.00 hours/sem.
Credit (sem. 1)	The preparation for a Credit: 10.00 hours/sem.	The written credit: 4.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	The grade from lectures is evaluated according to student's presence at lectures.
Class	The grade from classes, after passing all MEKs, is evaluated on basis tests and an activity in time of classes. This indirect result is recalculated to final grade according to the following scale: 100%-91% - 5.0, 90%-81% - 4.5, 80%-71% - 4.0, 70%-61% - 3.5, 60%-0% - 3.0.
The final grade	The final grade is the grade from the test.

Sample problems

Required during the exam/when receiving the credit
Semestr 1.pdf

Realized during classes/laboratories/projects
(-)

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: yes

1	L. Olszowy; T. Zając	On Darbo- and Sadovskii-Type Fixed Point Theorems in Banach Spaces	2024
2	S. Dudek; L. Olszowy	Measures of noncompactness in the space of regulated functions on an unbounded interval	2022
3	S. Dudek; L. Olszowy	Remarks on incorrect measure of noncompactness in BC (R+ x R+)	2022
4	J. Banaś; L. Olszowy	Remarks on the space of functions of bounded Wiener-Young variation	2020
5	L. Olszowy; T. Zając	Some inequalities and superposition operator in the space of regulated functions	2020
6	S. Dudek; L. Olszowy	Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval	2020
7	J. Banaś; L. Olszowy	On the equivalence of some concepts in the theory of Banach algebras	2019
8	L. Olszowy	Measures of noncompactness in the space of regulated functions	2019