Course: Functional Analysis 1

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Course title Functional Analysis 1
Course code KMA/FA1
Organizational form of instruction Lecture + Exercise
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 5
Language of instruction Czech
Status of course Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Staněk Svatoslav, prof. RNDr. CSc.
  • Vodák Rostislav, RNDr. Ph.D.
Course content
1. Metric spaces, complete metric spaces, compactness, continuous operators, Banach Fixed Point Theorem. 2. Linear spaces, basis, direct sum, linear operators and functionals, extension of linear functionals, projections, alpha and delta index of a linear operator. 3. Normed linear spaces, Banach spaces, complete hull, continuous linear operators, inverse operators. 4. The space of continuous linear operators, dual spaces, the Hahn-Banach Theorem and its consequences. 5. Canonical mappings, reflexive spaces. 6. Weak convergence. 7. Spaces with scalar products, Hilbert spaces, orthogonal systems, Fourier series, orthogonal projections, the Riesz Representation Theorem. 8. Adjoint operators. 9. Completely continuous linear mappings.

Learning activities and teaching methods
Monologic Lecture(Interpretation, Training), Dialogic Lecture (Discussion, Dialog, Brainstorming)
  • Attendace - 52 hours per semester
  • Homework for Teaching - 20 hours per semester
  • Preparation for the Course Credit - 20 hours per semester
  • Preparation for the Exam - 60 hours per semester
Learning outcomes
Master basic methods and tools of linear functional analysis.
Comprehension Understand the mathematical theory of linear operators in linear spaces.
Prerequisites
Understanding the basic elements of mathematical analysis including the mathematical tools of differential and integral calculus.

Assessment methods and criteria
Oral exam, Written exam

Credit: active participation, homework. Exam: written test, the student has to understand the subject and be able to prove principal results.
Recommended literature
  • A. N. Kolmogorov, S. V. Fomin. (1975). Základy teorie funkcí a funkcionální analýzy. SNTL, Praha.
  • Conway, J. B. (1990). A course in functional analysis. Springer.
  • J. Lukeš. (2001). Zápisky z funkcionální analýzy. MatFyzPress.
  • K. Najzar. (1988). Funkcionální analýza. SPN, Praha.
  • Taylor, A. E. (1977). Úvod do funkcionální analýzy. Academia, Praha.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Discrete Mathematics (2016) Mathematics courses 3 Winter