Course detail

Functional Analysis I

FSI-SU1Acad. year: 2024/2025

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.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. Mgr. Pavel Řehák, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Differential calculus, integral calculus, differential equations, linear algebra, elements of the set theory, elements of numerical mathematics.

Rules for evaluation and completion of the course

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.
The attendance in seminars will be checked. Students have to pass a test.

Aims

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)
A. Torchinsky, Problems in real and functional analysis, American Mathematical Society 2015. (EN)
C. Costara, D. Popa, Exercises in functional analysis, Kluwer 2003. (EN)
D. Farenick, Fundamentals of functional analysis, Springer 2016. (EN)
D. H. Griffel, Applied functional analysis, Dover 2002. (EN)
E. Zeidler, Applied functional analysis: Main principles and their applications, Springer, 1995. (EN)
F. Burk, Lebesgue measure and integration: An introduction, Wiley 1998. (EN)
I. Netuka, Základy moderní analýzy, MatfyzPress 2014. (CS)
J. Franců, Funkcionální analýza 1, FSI VUT 2014. (CS)
J. Lukeš, Zápisky z funkcionální analýzy, Karolinum 1998. (CS)
Z. Došlá, O. Došlý, Metrické prostory: teorie a příklady, PřF MU Brno 2006. (CS)

Recommended reading

A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)
D. Farenick, Fundamentals of functional analysis, Springer 2016. (EN)
D. H. Griffel, Applied functional analysis, Dover 2002. (EN)
I. Netuka, Základy moderní analýzy, MatfyzPress 2014. (CS)
J. Franců, Funkcionální analýza 1, FSI VUT 2014. (CS)

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 3 year of study, summer semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CLS , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. Mgr. Pavel Řehák, Ph.D.

Syllabus

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

26 hod., compulsory

Teacher / Lecturer

prof. Mgr. Pavel Řehák, Ph.D.

Syllabus

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.