Course detail
Functional Analysis I
FSI-SU1 Acad. year: 2021/2022 Summer semester
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, theory of linear functionals, 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.
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 a control
test 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
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 a test.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's
branch ---: no specialisation, 5 credits, compulsory
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.
Mappings between metric spaces. Banach fixed point theorem.
Applications. Precompact sets and relatively compact sets.
Arzelá-Aascoli theorem. Examples.
Elements of the theory of measure and integral
Motivation. Lebesgue measure. Measurable functions. Lebesgue integral.
Basic properties. Limit theorems. Lebesgue spaces. Examples.
Normed linear spaces
Basic concepts and facts. Banach spaces. Isometry. Homeomorphism.
Influence of the dimension of the space.
Infinite series in Banach spaces. The Schauder fixed point theorem and applications.
Examples.
Unitary spaces
Basic concepts and facts. Hilbert spaces. Isometry.
Orthogonality. Orthogonal projection. General Fourier series. Riesz-Fischer theorem.
Separable Hilbert spaces. Examples.
Linear functionals and operators, dual spaces
The concept of linear functional. Linear functionals in normed spaces
Continuous and bounded functionals. Hahn-Banach theorem and its consequences.
Dual spaces. Reflexive spaces.
Banach-Steinhaus theorem and its consequences. Weak convergence.
Examples
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.
Exercise
26 hours, compulsory
Teacher / Lecturer
Syllabus
Practising the subject-matter presented at the lectures mainly on particular examples of finite dimensional spaces, spaces of sequences and spaces of continuous and integrable functions.