Course detail
Functional Analysis and Function Spaces
FSI-9FAPAcad. year: 2024/2025
The course deals with basic topics of the functional analysis and function spaces and their application in analysis of probloms of mathematical physics.
Language of instruction
Czech
Mode of study
Not applicable.
Guarantor
Department
Entry knowledge
Differential and integral calculus, numerical methods, ordinary differential equations.
Rules for evaluation and completion of the course
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.
Absence has to be made up by self-study using lecture notes.
Absence has to be made up by self-study using lecture notes.
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.
Knowledge of basic topics of the metric, linear normed and unitary spaces,
Lebesgue integral and ability to apply this knowledge in practice.
Knowledge of basic topics of the metric, linear normed and unitary spaces,
Lebesgue integral and ability to apply this knowledge in practice.
Study aids
Not applicable.
Prerequisites and corequisites
Not applicable.
Basic literature
Kufner, A., John, O., Fučík, S.: Function spaces. Academia, Praha, 1977. (EN)
Nečas, J.: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012. (EN)
Rektorys, K.: Variační metody v inženýrských problémech a v problémech matematické fyziky. SNTL, Praha, 1974. (CS)
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)
Nečas, J.: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012. (EN)
Rektorys, K.: Variační metody v inženýrských problémech a v problémech matematické fyziky. SNTL, Praha, 1974. (CS)
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)
Recommended reading
Čech, E.: Bodové množiny, Academia, Praha, 1974, 288 stran
(CS)
Franců, J.: Funkcionální analýza 1, Akad. nakl. CERM, Brno 2014 (CS)
Kolmogorov, A. N., Fomin, S. V. : Základy teorie funkcí a funkcionální analýzy SNTL, Praha 1975. (CS)
Franců, J.: Funkcionální analýza 1, Akad. nakl. CERM, Brno 2014 (CS)
Kolmogorov, A. N., Fomin, S. V. : Základy teorie funkcí a funkcionální analýzy SNTL, Praha 1975. (CS)
Classification of course in study plans
Type of course unit
Lecture
20 hod., 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.
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.