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 discused 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: Requirements accordant with Rules and Regulations of studies

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, awaresness of the level of own knowledge and awaresness 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 OEK
01	student knows how to investigate the spectrum and he can calculate the resolvent of the simpliest operators	lectures, problem exercises	test, exam	K_W04+ K_W07+ K_U02+ K_U04+ K_U09++ K_U10++ K_U13+ K_U14+ K_U16+ K_K01+ K_K02+ K_K07+	X2A_W02 X2A_U01 X2A_U02 X2A_U03 X2A_U05 X2A_U06 X2A_U07 X2A_K01 X2A_K02 X2A_K06
02	student can calculate the norm of a linear bounded functional on some Banach space	lectures, problem exercises	test, exam	K_W01+ K_W02+ K_U05+ K_U07+ K_U09+ K_U10+ K_U13+ K_K04+ K_K07+	X2A_W03 X2A_U01 X2A_U02 X2A_U05 X2A_K03 X2A_K04 X2A_K06
03	student can calculate the adjoint (Hermitian adjoint) of some operators	lectures, problem exercises	test, exam	K_W01+ K_W02+ K_W03+ K_W05+ K_W07+ K_U01+ K_U03+ K_U09+++ K_U15+ K_K04+	X2A_W02 X2A_W03 X2A_U01 X2A_U02 X2A_U05 X2A_U06 X2A_U08 X2A_U09 X2A_K03 X2A_K04
04	student is able to examine the weak convergence in some Banach spaces	lectures, problem exercises	test, exam	K_W01+ K_W03+ K_U01+ K_U02+ K_U03+ K_U05++ K_U07+ K_U09+++ K_K02+	X2A_U01 X2A_U02 X2A_U03 X2A_U05 X2A_K01 X2A_K02

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.	W01-04, C01-04	MEK02 MEK03 MEK04
2	TK02	2. Operators in Hilbert spaces. Adjoint, Hermitian, unitary operators. 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.	W05-10, C05-09	MEK01 MEK02 MEK03
2	TK03	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. 5. Fixed point theorems. The Banach and Schauder theorems. Examples of applications in theory of differential and integral equations.	W11-15, C10-13	MEK02 MEK04
2	TK04	Tests based on the materials covered during lectures and tutorials.	C14-15	MEK01 MEK02 MEK03 MEK04

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: 5.00 hours/sem. Studying the recommended bibliography: 5.00 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: 30.00 hours/sem.	Finishing/Studying tasks: 5.00 hours/sem.
Advice (sem. 2)	The preparation for Advice: 4.00 hours/sem.	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: no