The aim of studying and bibliography

The main aim of study: To acquaint students with the basic sequence and function linear spaces occurring in functional analysis.

The general information about the module: The module is implemented in the fifth semester. It consists of 30 hours of lectures and 15 hours of classes. It ends with an exam.

others: Literatura wykorzystywana podczas zajęć zostanie podana poźniej.

Bibliography required to complete the module

Bibliography used during lectures

1	Banaszak G., Gajda W.	Elementy algebry liniowej	Wydawnictwa Naukowe-Techniczne, Warszawa.	2002
2	Gleichgewicht B.	Algebra	Oficyna Wydawnicza GiS, Wrocław.	2004
3	Lusternik L. A., Sobolew W. I.	Elementy analizy funkcjonalnej	Państwowe Wydawnictwo Naukowe, Warszawa.	1959
4	Musielak J.	Wstęp do analizy funkcjonalnej	Państwowe Wydawnictwo Naukowe, Warszawa.	1989

Bibliography used during classes/laboratories/others

1	Chmieliński J.	Analiza funkcjonalna. Notatki do wykładu	Wydawnictwo Naukowe Akademii Pedagogicznej w Krakowie.	2004
2	Prus S., Stachura A.	Analiza funkcjonalna w zadaniach	Wydawnictwo Naukowe PWN, Warszawa.	2007
3	Rutkowski J.	Algebra abstrakcyjna w zadaniach	Wydawnictwo Naukowe PWN, Warszawa.	2009

Bibliography to self-study

1

Musielak J.

Wstęp do analizy funkcjonalnej

Państwowe Wydawnictwo Naukowe, 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: The student is familiar with linear spaces and basic algebraic structures. The student knows the notions of a sequence, bounded sequence, convergent sequence, a function and bounded function.

Basic requirements in category skills: Students can check the linear independence of vectors and identify the bases of elementary linear spaces.

Basic requirements in category social competences: Student is prepared to undertake objective and justified actions in order to solve the posed exercise.

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	checks the properties of operations	lectures, exercises	written test	K_W02+ K_W04+ K_W06+ K_U01+	P6S_UK P6S_WG P6S_WK
02	checks whether or not a given structure is a ring and a field, checks if two given fields are isomorphic	lectures, exercises	written test	K_W05+ K_W06+ K_U01+	P6S_UK P6S_WG
03	checks the conditions required for the linear space, can verify if the subset of a linear space is a subspace	lectures, exercises	exam	K_W02+ K_W04++ K_W05++ K_U01+	P6S_UK P6S_WG P6S_WK
04	has a basic knowledge concerning linear spaces, both sequence and function, that are used in functional analysis	lectures, exercises	exam	K_W01+ K_W03+ K_W04+ K_W05+ K_K01+	P6S_KK P6S_WG P6S_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
5	TK01	Revision of the knowledge concerning general algebra. Internal and external operation. Definition of a group, properties of a group. Subgroup. Examples of the groups and subgroups. Homomorphism of groups. Isomorphic groups.	W1-W4, C1-C2	MEK01
5	TK02	Revision of the knowledge concerning general algebra. Definition and examples of the rings. Special elements in the rings.	W5-W6, C3	MEK01 MEK02
5	TK03	Field in comparison with ring. Field of real numbers and field of complex numbers. The other examples of fields. Homomorphism and isomorphism of fields.	W7-W10, C4-C6	MEK01 MEK02
5	TK04	Revision of the knowledge concernig linear algebra. Definition of linear space. n-dimensional linear space and infinite dimensional linear space. The subspace of the linear space.	W11-W12, C7	MEK01 MEK03
5	TK05	Linear space of functions defined on a given set with the values ​​in fixed field.	W13-W14, C8	MEK03
5	TK06	R∞ space as a sequential space. Linear space C∞ of the sequences of complex numbers.	W15-W16, C9	MEK03 MEK04
5	TK07	Linear space m of the bounded sequences of real or complex terms. The space c of the convergent sequences. The space c_0 of the sequences convergent to zero.	W17-W20, C10	MEK03 MEK04
5	TK08	Holder's inequality. Minkowski inequality for the sums. Minkowski inequality for the series.	W21-W22	MEK03
5	TK09	Sequential space l^p of the convergent series with the p-th power.	W23-W24, C11-C12	MEK03 MEK04
5	TK10	Llinear space of the real-valued functions defined on the interval [a,b​​]. Linear space of the real-valued functions defined and continuous on the interval [a,b​​].	W25-W26, C13-C14	MEK03 MEK04
5	TK11	Linear space R^p of the p-th power-integrable functions in the Riemann sense.	W27-W30, C15	MEK03 MEK04

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 5)		contact hours: 30.00 hours/sem.	complementing/reading through notes: 10.00 hours/sem.
Class (sem. 5)	The preparation for a Class: 15.00 hours/sem. The preparation for a test: 10.00 hours/sem.	contact hours: 15.00 hours/sem.	Finishing/Studying tasks: 10.00 hours/sem.
Advice (sem. 5)
Exam (sem. 5)	The preparation for an Exam: 15.00 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	Credit for the lecture is based on attendance.
Class	A credit for the classes is based on the result of test.
The final grade	The final grade is the arithmetic mean of grades from classes and exam arounded to applicable scale

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