Course title  Introduction to Functional Analysis 

Course code  UMB/580 
Organizational form of instruction  Lecture 
Level of course  Bachelor 
Year of study  not specified 
Frequency of the course  In each academic year, in the winter semester. 
Semester  Winter 
Number of ECTS credits  5 
Language of instruction  Czech 
Status of course  Compulsoryoptional 
Form of instruction  Facetoface 
Work placements  This is not an internship 
Recommended optional programme components  None 
Lecturer(s) 


Course content 
Sigma algebra, introduction of a Lebesgue measure, measurable sets, measurable functions. Definition of the Lebesgue integral  fundamental properties. Convergence theorems. Metric spaces, normed linear spaces, Banach and Hilbert spaces. Spaces of smooth functions, Lp spaces. Linear operators, continuous operators.

Learning activities and teaching methods 
Monologic (reading, lecture, briefing), Demonstration

Learning outcomes 
Students should acquaint themselves with Lebesgue integral and basic functional spaces.
We introduce sigma algebra, outer measure and then Lebesgue measure, measure sets and functions. We define Lebesgue integral and show its basic properties. We work with metric spaces, normed spaces, in particular Banach and Hilbert spaces. As examples we will use spaces of smooth functions, Lp spaces and Sobolev spaces. We will study linear and continuous operators and their properties. We show a power and application of fixed point theorems. 
Prerequisites 
We assume knowledge of differentail and integral calculus (UMB/564, UMB/565, UMB/566) and Linear algebra UMB/551. It is better to pass before also Linear algebra II UMB/585, where similar topics is studied in spaces of finite dimension.
UMB/CV566  or  UMB/CV585  or  UMB/566  or  UMB/566K  or  UMB/585 
Assessment methods and criteria 
Oral examination
Active participation on lectures, solving of simple homeworks. Final oral exam. 
Recommended literature 

Study plans that include the course 
Faculty  Study plan (Version)  Category of Branch/Specialization  Recommended semester  

Faculty: Faculty of Science  Study plan (Version): Secondary Schools Teacher Training in Mathematics (2012)  Category: Pedagogy, teacher training and social care    Recommended year of study:, Recommended semester: Winter 
Faculty: Faculty of Science  Study plan (Version): Applied Mathematics (2010)  Category: Mathematics courses  3  Recommended year of study:3, Recommended semester: Winter 
Faculty: Faculty of Science  Study plan (Version): Secondary Schools Teacher Training in Mathematics (1)  Category: Pedagogy, teacher training and social care    Recommended year of study:, Recommended semester: Winter 
Faculty: Faculty of Science  Study plan (Version): Secondary Schools Teacher Training in Mathematics (1)  Category: Pedagogy, teacher training and social care    Recommended year of study:, Recommended semester: Winter 
Faculty: Faculty of Science  Study plan (Version): Secondary Schools Teacher Training in Mathematics (1)  Category: Pedagogy, teacher training and social care    Recommended year of study:, Recommended semester: Winter 