Functional Analysis and Function Spaces (FSI-9FAP)

Academic year 2022/2023
Supervisor: prof. RNDr. Jan Franců, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Course contents:
The course deals with basic topics of the functional analysis and function spaces and their application in analysis of probloms of mathematical physics.
Teaching methods and criteria:
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.
Controlled participation in lessons:
Absence has to be made up by self-study using lecture notes.
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 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.
Literature - fundamental:
1. Rektorys, K.: Variační metody v inženýrských problémech a v problémech matematické fyziky. SNTL, Praha, 1974.
2. Kufner, A., John, O., Fučík, S.: Function spaces. Academia, Praha, 1977.
3. Nečas, J.: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012.
4. Yosida, K. : Functional analysis, Springer, Berlin, 1965
5. Ženíšek, A.: Nonlinear elliptic and evolution problems and their finite element approximations. Academic Press, London, 1990.
Literature - recommended:
1. Franců, J.: Funkcionální analýza 1, Akad. nakl. CERM, Brno 2014
2. Čech, E.: Bodové množiny, Academia, Praha, 1974, 288 stran
3. Kolmogorov, A. N., Fomin, S. V. : Základy teorie funkcí a funkcionální analýzy SNTL, Praha 1975.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D-APM-K combined study --- -- DrEx 0 Recommended course 3 1 S
D-APM-P full-time study --- -- DrEx 0 Recommended course 3 1 S