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Course info
KMA / FA
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Course description
Department/Unit / Abbreviation
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KMA
/
FA
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Functional Analysis
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
5
Cred.
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Type of completion
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Written
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Type of completion
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Written
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Time requirements
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Lecture
4
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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No
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Included in study average
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YES
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Language of instruction
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Czech
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Occ/max
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Automatic acceptance of credit before examination
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No
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Summer semester
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2 / -
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0 / -
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0 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Winter + Summer
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Semester taught
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Winter + Summer
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Minimum (B + C) students
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1
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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Czech
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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KMA/MMA
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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The goal of the subject is an introduction of students to the basic ideas and methods of functional analysis.
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Requirements on student
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Before the exam the student has to solve given written work.
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Content
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1. Spectral theory of linear operators
2. Differential calculus in Banach spaces
3. Abstract implicit function theorem
4. Theory of monotone operators
5. Critical points of functionals
6. Analysis of noncoercive operators
7. Brouwer and Leray-Schauder degree
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Activities
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Fields of study
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Na uvedené stránce je k dispozici odkaz na záznam online přednášek.
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Guarantors and lecturers
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Literature
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Graduate study programme term essay (40-50)
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50
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Contact hours
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65
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Preparation for an examination (30-60)
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40
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Total
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155
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
definovat a na příkladech vysvětlit pojmy otevřená, uzavřená množina, úplný prostor, kompaktní prostor, separabilní prostor, Banachův a Hilbertův prostor |
definovat a uvést příklady kompaktního, duálního a samoadjungovaného operátoru |
definovat duální prostor, reflexivní prostor, slabou konvergenci a zformulovat Eberleinovu-Šmuljanovu větu |
definovat metrický, normovaný, unitární prostor a uvést jejich příklady |
zavést ortonormální systém a Fourierovu řadu na Hilbertově prostoru |
zavést prostor spojitě diferencovatelných funkcí, prostor spojitých funkcí s kompaktním nosičem, Sobolevovy prostory |
zformulovat Rieszovu větu o reprezentaci spojitého lineárního funkcionálu |
zformulovat větu o minimu kvadratického funkcionálu |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
dokázat, že vlastní čísla symetrického operátoru na Hilbertově prostoru jsou reálná a odpovídající vlastní vektory jsou kolmé |
formulovat, dokázat a aplikovat Banachovu větu o kontrakci |
zformulovat a dokázat Minkowského a Hölderovu nerovnost |
zformulovat a dokázat Schwarzovu nerovnost |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
definovat pojem spektra lineárního operátoru |
definovat základní pojmy differenciálního počtu v Banachových prostorech |
zformulovat abstraktní větu o implicitní funkci a vysvětlit její aplikace |
definovat kritický bod, bod minima a maxima funkcionálu |
definovat pojem motónního a koercitivního operátoru |
Skills - skills resulting from the course: |
stanovit spektrum daného operátoru |
vypočítat Fréchetovu a Gateauxovu derivaci daného zobrazení a funkcionálu |
pro zadaný funkcionál stanovit kritické body a rozhodnout, zda se jedná o bod minima nebo maxima |
aplikovat teorii motónních a nekoercitivních operátorů na praktické úlohy v oblasti ODR a PDR |
vysvětlit pojem Brouwerův stupeň a Leray-Schauderův stupeň zobrazení |
Competences - competences resulting from the course: |
N/A |
N/A |
N/A |
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Combined exam |
Skills - skills achieved by taking this course are verified by the following means: |
Oral exam |
Skills demonstration during practicum |
Competences - competence achieved by taking this course are verified by the following means: |
Oral exam |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Lecture |
Individual study |
Skills - the following training methods are used to achieve the required skills: |
Practicum |
Task-based study method |
Competences - the following training methods are used to achieve the required competences: |
Task-based study method |
Self-study of literature |
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