Course detail

# Functional Analysis I

FSI-SU1Acad. year: 2017/2018

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.

Supervisor

Department

Learning outcomes of the course unit

Basic knowledge of metric, linear, normed and unitary spaces, elements of Lebesgue integral 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.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

F. Burk, Lebesgue measure and integration: An introduction, Wiley 1998. (EN)

C. Costara, D. Popa, Exercises in functional analysis, Kluwer 2003. (EN)

Z. Došlá, O. Došlý, Metrické prostory: teorie a příklady, PřF MU Brno 2006. (CS)

J. Franců, Funkcionální analýza 1, FSI VUT 2014. (CS)

D. H. Griffel, Applied functional analysis, Dover 2002. (EN)

A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)

E. Kreyszig, Introductory functional analysis with applications, J. Wiley 1978. (EN)

J. Lukeš, Zápisky z funkcionální analýzy, Karolinum 1998. (CS)

B. Rynne, M. Youngson, Linear functional analysis, Springer 2008. (EN)

K. Saxe, Beginning functional analysis, Springer 2002. (EN)

A. E. Taylor, Úvod do funkcionální analýzy, Academia, Praha 1973. (CS)

A. Torchinsky, Problems in real and functional analysis, American Mathematical Society 2015. (EN)

E. Zeidler, Applied functional analysis: Main principles and their applications, Springer, 1995. (EN)

Planned learning activities and teaching methods

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 two control tests 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.

Language of instruction

Czech

Work placements

Not applicable.

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.

Specification of controlled education, way of implementation and compensation for absences

The attendance in seminars will be checked. Students have to pass two tests.

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

Metric spaces

Basic concepts and facts. Examples. Closed and open sets. Convergence. Separable metric spaces. Complete metric spaces. Compact spaces. Mappings between metric spaces. Banach fixed point theorem.

Elements of the theory of measure and integral

Lebesgue measure. Measurable functions. Lebesgue integral. Limit theorems.

Normed linear spaces

Basic concepts and facts. Examples. Finite vs. infinite dimension. Banach spaces. Examples. (Relative) compactness. Arzelá-Ascoli theorem. The Schauder fixed point theorem. Applications.

Unitary spaces

Basic concepts and facts. Hilbert spaces. Examples. Finite vs. infinite dimension. Orthogonality. General Fourier series. Riesz-Fischer theorem.

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.

Linear functionals and operators, dual spaces and operators

Space of linear operators. Continuity. Boundedness. Invertibility. Influence of the dimension of the space. Dual spaces. Reflexive space. Weak convergence. Dual and adjoint operators. Hahn-Banach theorem and its consequences. Banach-Steinhaus theorem and its consequences.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

Practising the subject-matter presented at the lectures on particular examples of finite dimensional spaces, spaces of sequences and spaces of continuous and integrable functions.

eLearning

**eLearning:** currently opened course