Course detail

Functional Analysis and Function Spaces

FSI-9FAP Acad. year: 2024/2025 Summer semester

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

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.

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.

The study programmes with the given course

Programme D-APM-K: Applied Mathematics, Doctoral, recommended course

Programme D-APM-P: Applied Mathematics, Doctoral, recommended course

Type of course unit

 

Lecture

20 hours, optionally

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.