Course detail
Functional Analysis I
FSI-SU1Acad. year: 2017/2018
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
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. 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)
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
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. Compact spaces. Mappings between metric spaces. Banach fixed point theorem.
Elements of the theory of measure and integral
Lebesgue measure. Measurable functions. Lebesgue integral. Limit theorems.
Normed linear spaces
Basic concepts and facts. Examples. Finite vs. infinite dimension. Banach spaces. Examples. (Relative) compactness. Arzelá-Ascoli theorem. The Schauder fixed point theorem. Applications.
Unitary spaces
Basic concepts and facts. Hilbert spaces. Examples. Finite vs. infinite dimension. Orthogonality. General Fourier series. Riesz-Fischer theorem.
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.
Linear functionals and operators, dual spaces and operators
Space of linear operators. Continuity. Boundedness. Invertibility. Influence of the dimension of the space. Dual spaces. Reflexive space. Weak convergence. Dual and adjoint operators. Hahn-Banach theorem and its consequences. Banach-Steinhaus theorem and its consequences.
Exercise
Teacher / Lecturer
Syllabus