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: Norms of operator and functional, dual space. Hahn-Banach theorem. Dual spaces of classical sequence and function Banach spaces. Riesz theorem. The second dual space. Reflexivity. Adjoint, Hermitian, unitary operators. Eigenvalues, eigenvectors, spectrum, set of resolvent, resolvent of operator, von Neumann series. Integral operators, the Fredholm integral equations. The spectral theorem for compact operators. Locally convex spaces. Hyperplane separation theorem for convex sets. The Mazur, Alaoglu, Goldstine, Eberlein theorems. Theorems for reflexive spaces. The Banach and Schauder fixed point theorems. Examples of applications in theory of differential and integral equations.

Bibliography required to complete the module

Bibliography used during lectures

1	J. Musielak	Wstęp do analizy funkcjonalnej	wyd. 2 poprawione, PWN, Warszawa.	1989
2	W. Rudin	Analiza funkcjonalna	wyd. 2, Wydawnictwo Naukowe PWN, Warszawa.	2002

Bibliography used during classes/laboratories/others

1

S. Prus, A.Stachura

Analiza funkcjonalna w zadaniach

Wydawnictwo Naukowe PWN, Warszawa.

2007

Bibliography to self-study

1

J. Musielak

Wstęp do analizy funcjonalnej

wyd. 2 poprawione, PWN, Warszawa.

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, functional analysis I

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 knows how to investigate the spectrum and he can calculate the resolvent of the simplest operators	lectures, problem exercises	test, exam	K_W01+ K_W04++ K_W05++ K_W07+++ K_U04+ K_U05+ K_U09+ K_U10++ K_U13+ K_U15+ K_U16+ K_K02+	P7S_KK P7S_KO P7S_UK P7S_UO P7S_UU P7S_UW P7S_WG
02	student can calculate the norm of a linear bounded functional on some Banach space	lectures, problem exercises	test, exam	K_W04+ K_W05+ K_U02+ K_U05+ K_U07++ K_K01+	P7S_KK P7S_UK P7S_UO P7S_UW P7S_WG
03	student can calculate the adjoint (Hermitian adjoint) of some operators	lectures, problem exercises	test, exam	K_W02+ K_W04++ K_W05++ K_W07++ K_U05+ K_U09++ K_U10+ K_U13+ K_U16+ K_K07+	P7S_KK P7S_KO P7S_KR P7S_UK P7S_UU P7S_UW P7S_WG P7S_WK
04	student is able to examine the weak convergence in some Banach spaces	lectures, problem exercises	test, exam	K_W01+ K_W02++ K_W03+ K_U01+ K_U02+ K_U03++ K_U07++ K_U09++ K_U14+ K_K04+	P7S_KO P7S_KR P7S_UK P7S_UO P7S_UW P7S_WG P7S_WK

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
2	TK01	1. Continuous linear functionals. Norms of functional, dual space. Hahn-Banach theorem. Dual spaces of classical sequence and function Banach spaces. Riesz theorem. The second dual space. Reflexivity.	W01-08, C01-12	MEK02 MEK03 MEK04
2	TK02	2. Operators in Hilbert spaces. Adjoint, Hermitian, unitary operators.	W09-14, C13-20	MEK01 MEK02 MEK03
2	TK03	3. Elements of spectral analysis. Eigenvalues, eigenvectors, spectrum, set of resolvent, resolvent of operator, von Neumann series. Integral operators, the Fredholm integral equations. The spectral theorem for compact operators. Fixed point theorems. The Banach and Schauder theorems. Examples of applications in theory of differential and integral equations.	W15-24, C21-32	MEK01 MEK02 MEK04
2	TK04	4. Weak convergence and weak topologies in Banach spaces. Locally convex spaces. Hyperplane separation theorem for convex sets. The Mazur, Alaoglu, Goldstine, Eberlein theorems. Theorems for reflexive spaces.	W25-30, C33-39	MEK02 MEK03 MEK04
2	TK05	Tests based on the materials covered during lectures and tutorials.	C40-45

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 2)		contact hours: 30.00 hours/sem.	complementing/reading through notes: 2.00 hours/sem. Studying the recommended bibliography: 10.00 hours/sem.
Class (sem. 2)	The preparation for a Class: 30.00 hours/sem. The preparation for a test: 10.00 hours/sem.	contact hours: 45.00 hours/sem.	Finishing/Studying tasks: 5.00 hours/sem.
Advice (sem. 2)		The participation in Advice: 4.00 hours/sem.
Exam (sem. 2)	The preparation for an Exam: 10.00 hours/sem.	The written exam: 3.00 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	A credit for the lecture is based on the result of the written exam. The written exam consists of theory questions and problems on topics discussed in lectures and tutorials.
Class	A credit for the tutorials is based on the results of tests and oral answers.
The final grade	After the credit of all types of classes, the final grade is the average of grade of classes and grade of exam (by a condition that student has passed the exam).

Sample problems

Required during the exam/when receiving the credit
(-)

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