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Course, academic year 2023/2024
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Mathematical analysis VI - NMUM402
Title: Matematická analýza VI
Guaranteed by: Department of Mathematics Education (32-KDM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. RNDr. Ivan Netuka, DrSc.
doc. RNDr. Antonín Slavík, Ph.D.
Incompatibility : NMTM402
Interchangeability : NMTM402
Is incompatible with: NMTM402
Is interchangeable with: NMTM402
Annotation -
Last update: T_KDM (12.04.2016)
Basic course in mathematical analysis (Fourier series, metric spaces, normed linear spaces).
Course completion requirements -
Last update: doc. RNDr. Antonín Slavík, Ph.D. (29.04.2020)

It is necessary to pass two written tests during the term. If necessary, the second test might be assigned online.

Literature -
Last update: T_KDM (12.04.2016)

A. Pinkus, S. Zafrany: Fourier Series and Integral Transforms. Cambridge University Press, 1997.

J. Muscat: Functional Analysis. An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras. Springer, 2014.

W. A. Sutherland: Introduction to Metric and Topological Spaces (Second Edition). Oxford University Press, 2009.

W. Rudin: Principles of mathematical analysis. McGraw-Hill, Inc., New York, 1976.

Requirements to the exam -
Last update: doc. RNDr. Antonín Slavík, Ph.D. (28.10.2019)

An oral exam following the syllabus of the subject in the scope of the lecture.

Syllabus -
Last update: T_KDM (12.04.2016)
  • Fourier series: Trigonometric polynomial, trigonometric series, uniform convergence of trigonometric series, Fourier coefficients. Orthonormal set in L^2, Bessel's inequality, Riesz-Fischer theorem, complete orthonormal set, Parseval's identity. Riemann-Lebesgue lemma. Pointwise convergence of Fourier series. Parseval's identity for trigonometric system, completeness of trigonometric system.
  • Metric spaces, normed linear spaces: Metric, metric space, subspace, examples; norm, normed linear space, examples. Basic concepts in metric spaces: open and closed set; properties of the system of open and closed sets; topology, topological space; accumulation point, isolated point, closure, interior, diameter of a set.
  • Continuous mappings, convergence: Continuous mapping at a point (metric spaces). Continuity and preimages of open set; convergent sequence in a metric space, uniqueness of a limit, characterization of continuity using sequences.
  • Complete metric spaces: Cauchy sequence, complete space, examples. Subset of a complete space is complete if and only if it is closed. Cantor's intersection theorem. Contraction mapping, Banach fixed point theorem.

 
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