Course detail

Functional Analysis and Function Spaces

FSI-9FAPAcad. year: 2020/2021

The course deals with basic topics of the functional analysis and function spaces and their application in analysis of probloms of mathematical physics.

Learning outcomes of the course unit

Knowledge of basic topics of the metric, linear normed and unitary spaces,
Lebesgue integral and ability to apply this knowledge in practice.

Prerequisites

Differential and integral calculus, numerical methods, ordinary differential equations.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Franců, J.: Funkcionální analýza 1, Akad. nakl. CERM, Brno 2014 (CS)
Rektorys, K.: Variační metody v inženýrských problémech a v problémech matematické fyziky. SNTL, Praha, 1974. (CS)
Kufner, A., John, O., Fučík, S.: Function spaces. Academia, Praha, 1977. (EN)
Čech, E.: Bodové množiny, Academia, Praha, 1974, 288 stran (CS)
Kolmogorov, A. N., Fomin, S. V. : Základy teorie funkcí a funkcionální analýzy SNTL, Praha 1975. (CS)
Nečas, J.: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012. (EN)
Yosida, K. : Functional analysis, Springer, Berlin, 1965 (EN)
Ženíšek, A.: Nonlinear elliptic and evolution problems and their finite element approximations. Academic Press, London, 1990. (EN)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

Examination has a practical and a theoretical part. In the practical part student has to illustrate the given topics on particular examples. Theoretical part includes questions related to the subject-matter presented at the lectures.

Language of instruction

Czech, English

Work placements

Not applicable.

Aims

The aim of the course is to familiarise students with basic topics of the functional analysis and function spaces theory and their application to analysis of problems of mathematical physics.

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

Absence has to be made up by self-study using lecture notes.

Classification of course in study plans

  • Programme D-APM-K Doctoral, 1. year of study, summer semester, 0 credits, recommended

  • Programme D4P-P Doctoral

    branch D-APM , 1. year of study, summer semester, 0 credits, recommended

Type of course unit

 

Lecture

20 hours, optionally

Teacher / Lecturer

Syllabus

1 Metric and metric spaces, examples.
2 Linear and normed linear spaces, Banach spaces.
3 Scalar product and Hilbert spaces.
4 Examples of spaces: R^n, C^n, sequential spaces, spaces of continuous and integrable functions.
5 Elements of Lebesgue integral, Lebesgue spaces.
6 Generalized derivations, Sobolev spaces.
7 Traces. Theorem on traces.
8 Imbedding theorems. Density theorem.
9 Lax-Milgram lemma and its application to solvability if differential equations.
10 Relation between differential and integral equations.