Course detail
Functional Analysis I
FSI-SU1Acad. year: 2022/2023
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.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Learning outcomes of the course unit
Prerequisites
Co-requisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
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.
Course curriculum
Work placements
Aims
Specification of controlled education, way of implementation and compensation for absences
Recommended optional programme components
Prerequisites and corequisites
Basic literature
A. Torchinsky, Problems in real and functional analysis, American Mathematical Society 2015. (EN)
C. Costara, D. Popa, Exercises in functional analysis, Kluwer 2003. (EN)
D. Farenick, Fundamentals of functional analysis, Springer 2016. (EN)
D. H. Griffel, Applied functional analysis, Dover 2002. (EN)
E. Zeidler, Applied functional analysis: Main principles and their applications, Springer, 1995. (EN)
F. Burk, Lebesgue measure and integration: An introduction, Wiley 1998. (EN)
I. Netuka, Základy moderní analýzy, MatfyzPress 2014. (CS)
J. Franců, Funkcionální analýza 1, FSI VUT 2014. (CS)
J. Lukeš, Zápisky z funkcionální analýzy, Karolinum 1998. (CS)
Z. Došlá, O. Došlý, Metrické prostory: teorie a příklady, PřF MU Brno 2006. (CS)
Recommended reading
D. Farenick, Fundamentals of functional analysis, Springer 2016. (EN)
D. H. Griffel, Applied functional analysis, Dover 2002. (EN)
I. Netuka, Základy moderní analýzy, MatfyzPress 2014. (CS)
J. Franců, Funkcionální analýza 1, FSI VUT 2014. (CS)
Elearning
Classification of course in study plans
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
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
Teacher / Lecturer
Syllabus
Elearning