714-0522/01 – Functional Analysis (FA)
Gurantor department | Department of Mathematics and Descriptive Geometry | Credits | 4 |
Subject guarantor | doc. RNDr. Jaroslav Vlček, CSc. | Subject version guarantor | doc. RNDr. Jaroslav Vlček, CSc. |
Study level | undergraduate or graduate | Requirement | Choice-compulsory |
Year | 4 | Semester | winter |
| | Study language | Czech |
Year of introduction | 1998/1999 | Year of cancellation | 2015/2016 |
Intended for the faculties | HGF | Intended for study types | Follow-up Master |
Subject aims expressed by acquired skills and competences
- an obtaining of basic orientation in functional spaces,
- skilled use of functional analysis theory in formulation of applied problems.
Teaching methods
Lectures
Individual consultations
Tutorials
Summary
Metric spaces, linear spaces, normed linear spaces, orthogonality.
Cauchy's sequences, completeness.
Banach and Hilbert spaces.
Hilbertovy a Banachovy prostory.
Functionals and operators in Hilbert space. Schauder's theorem.
Compulsory literature:
REKTORYS, K.: Variational Methods in Mathematics, Science and
Engineering. SNTL, Praha, 1980
Recommended literature:
E. Zeidler: Applied Functional Analysis. Springer-Verlag, New York, 1995
Way of continuous check of knowledge in the course of semester
Cvičení (10-30 bodů):
- aktivní účast ve výuce (20% neúčasti tolerováno) ... 10 b.
- testy ... 20 b.
Zkouška (0-70 b.):
- praktická (písemná část) ... 50 b.
- teoretická část ... 20 b.
1. Metrický prostor. Základní metrické a topologické pojmy.
2. Separabilita a kompaktnost. Cauchyovské posloupnosti. Úplnost.
3. Lineární prostor. Definice a základní vlastnosti. Konvexnost.
4. Normovaný lineární prostor. Norma. Metrika indukovaná normou.
5. Banachův prostor. Věta o pevném bodě.
6. Unitární prostor. Skalární součin. Metrika indukovaná skalárním součinem.
7. Hilbertův prostor. Rieszova věta o reprezentaci.
8. Ortogonální systémy. Příklady.
9. Lineární funkcionály a operátory v Hilbertově prostoru.
E-learning
Other requirements
The other demands are not prescribed.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
1. Metric spaces. Separability and compactness. Cauchy sequences. Completness.
2. Linear space. Definition and basic properties. Convexity. Examples of linear spaces.
3. Norm. Metric induced by norm. Banach space.
4. Unitary spaces. Scalar product. Metric indued by scalar product. Hilbert space. Riesz theorem.
5. Ortogonal systems.
6. Linear operstors in Hilbert space.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction
Předmět neobsahuje žádné hodnocení.