Course detail
Functional Analysis I
FSI-SU1 Acad. year: 2016/2017 Summer semester
The course deals with basic topics of the functional analysis and their illustration on particular metric, linear normed and unitary spaces. Lebesgue measure and Lebesgue integral are also introduced. The results are applied to solving of problems of mathematical and numerical analysis.
Language of instruction
Czech
Number of ECTS credits
5
Supervisor
Department
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.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on 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 and passed the control test.
Examination has a practical and a theoretical part. In the practical part student has to illustrate the given tasks on particular examples.
Theoretical part includes questions related to the subject-matter presented at the lectures.
Aims
The aim of the course is to familiarise students with basic topics and procedures of the functional analysis used in other mathematical subjects.
Specification of controlled education, way of implementation and compensation for absences
Absence has to be made up by self-study using recommended literature.
The study programmes with the given course
Programme B3A-P: Applied Sciences in Engineering, Bachelor's
branch B-MAI: Mathematical Engineering, compulsory
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
Syllabus
1. Metric spaces – definition and examples, classification of subsets, separability, covergence, completness, compactness, compactness of sets in particular spaces.
2. Measura theory – Lebesgue measure, measurable functions, Lebesgue integral, Lebesgue dominant theorem.
3. Linear spaces – definition and examples, normed space, Euclidian space, Bessel inequality, Riesz-Fischer theorem, Hilbert space, characteristic property of Euclidian spaces.
4. Functionals – definition and examples, geometric interpretation, convex sets, convex functionals, Hahn-Banach theorem, continuous linear functionals, Hahn-Banach theorem in normed spaces.
5. Adjoint spaces – definition and examples, second adjoint spaces, weak convergence, Banach-Steinhaus theorem, weak convergence and bounded sets in adjoint spaces.
Exercise
26 hours, compulsory
Teacher / Lecturer
Syllabus
Practising the subject-matter presented at the lectures on particular examples of finite dimensional spaces, spaces of sequences and spaces of continuous and integrable functions.