Functional Analysis I (FSI-SU1)

Academic year 2021/2022
Supervisor: prof. Mgr. Pavel Řehák, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The aim of the course is to familiarise students with basic topics and procedures of functional analysis, which can be used in other branches of mathematics.
Learning outcomes and competences:
Basic knowledge of metric, linear, normed and unitary spaces, elements of Lebesgue integral, theory of linear functionals, and related concepts. Ability to apply these knowledges in practice.
Prerequisites:
Differential calculus, integral calculus, differential equations, linear algebra, elements of the set theory, elements of numerical mathematics.
Course contents:
The course deals with basic concepts and principles of functional analysis concerning, in particular, metric, linear normed and unitary spaces. Elements of the theory of Lebesgue measure and Lebesgue integral will also be mentioned. It will be shown how the results are applied in solving problems of mathematical analysis and numerical mathematics.
Teaching methods and criteria:
The course is taught through lectures explaining theoretical backgroung and basic principles of the discipline. Exercises are focused on managing practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
Course-unit credit is awarded on condition of having attended the
seminars actively (the attendance is compulsory) and passed a control
test during the semester.
Examination: It has oral form. Theory
as well as examples will be discussed. Students should show they are
familiar with basic topics and principles of the discipline and they
are able to illustrate the theory in particular situations.
Controlled participation in lessons:
The attendance in seminars will be checked. Students have to pass a test.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture Metric spaces
Basic concepts and facts. Examples. Closed and open sets.
Convergence. Separable metric spaces. Complete metric spaces.
Mappings between metric spaces. Banach fixed point theorem.
Applications. Precompact sets and relatively compact sets.
Arzelá-Aascoli theorem. Examples.

Elements of the theory of measure and integral
Motivation. Lebesgue measure. Measurable functions. Lebesgue integral.
Basic properties. Limit theorems. Lebesgue spaces. Examples.

Normed linear spaces
Basic concepts and facts. Banach spaces. Isometry. Homeomorphism.
Influence of the dimension of the space.
Infinite series in Banach spaces. The Schauder fixed point theorem and applications.
Examples.

Unitary spaces
Basic concepts and facts. Hilbert spaces. Isometry.
Orthogonality. Orthogonal projection. General Fourier series. Riesz-Fischer theorem.
Separable Hilbert spaces. Examples.

Linear functionals and operators, dual spaces
The concept of linear functional. Linear functionals in normed spaces
Continuous and bounded functionals. Hahn-Banach theorem and its consequences.
Dual spaces. Reflexive spaces.
Banach-Steinhaus theorem and its consequences. Weak convergence.
Examples

Particular types of spaces (in the framework of the theory under consideration).
In particular, spaces of sequences, spaces of continuous functions,
and spaces of integrable functions. Some inequalities.
    Exercise Practising the subject-matter presented at the lectures mainly on particular examples of finite dimensional spaces, spaces of sequences and spaces of continuous and integrable functions.
Literature - fundamental:
1. F. Burk, Lebesgue measure and integration: An introduction, Wiley 1998.
2. C. Costara, D. Popa, Exercises in functional analysis, Kluwer 2003.
3. Z. Došlá, O. Došlý, Metrické prostory: teorie a příklady, PřF MU Brno 2006.
4. D. Farenick, Fundamentals of functional analysis, Springer 2016.
5. J. Franců, Funkcionální analýza 1, FSI VUT 2014.
6. D. H. Griffel, Applied functional analysis, Dover 2002.
7. A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.
9. J. Lukeš, Zápisky z funkcionální analýzy, Karolinum 1998.
10. I. Netuka, Základy moderní analýzy, MatfyzPress 2014.
14. A. Torchinsky, Problems in real and functional analysis, American Mathematical Society 2015.
15. E. Zeidler, Applied functional analysis: Main principles and their applications, Springer, 1995.
Literature - recommended:
1. A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.
2. I. Netuka, Základy moderní analýzy, MatfyzPress 2014.
3. J. Franců, Funkcionální analýza 1, FSI VUT 2014.
4. D. Farenick, Fundamentals of functional analysis, Springer 2016.
5. D. H. Griffel, Applied functional analysis, Dover 2002.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-MAI-P full-time study --- no specialisation -- Cr,Ex 5 Compulsory 1 2 S