Theory of Real Functions 1 - NMMA403
Title: Reálné funkce 1
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2024
Semester: winter
E-Credits: 4
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. RNDr. Miroslav Zelený, Ph.D.
Teacher(s): doc. RNDr. Miroslav Zelený, Ph.D.
Class: M Mgr. MA
M Mgr. MA > Povinné
Classification: Mathematics > Real and Complex Analysis
Incompatibility : NRFA014
Interchangeability : NRFA014
Is interchangeable with: NRFA014
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Annotation -
Mandatory course for the master study branch Mathematical analysis. Recommended for the first year of master studies. Content: differentiation of measures, absolutely continuous functions, fuctions of bounded variation, Lipschitz function, Hausdorff measure and dimension.
Last update: T_KMA (10.05.2013)
Course completion requirements -

The exam is oral. The required knowledge corresponds to the sylabus at the presented extent.

Last update: Malý Jan, prof. RNDr., DrSc. (29.10.2019)
Literature -

L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.

W. Rudin, Real and Complex Analysis, Third edition. McGraw-Hill Book Co., New York, 1987.

Last update: T_KMA (02.05.2013)
Requirements to the exam -

The exam is oral. The required knowledge corresponds to the sylabus at the presented extent.

Last update: Malý Jan, prof. RNDr., DrSc. (29.10.2019)
Syllabus -

1. Differentiation of measures

  • covering theorems (Vitali, Besikovich, perhaps also Whitney)
  • maximal operator
  • application to absolutely continuous functions and to functions of bounded variation
  • mutual differentiation of two Radon measures
  • Lebesgue points of locally integrable functions
  • Rademacher theorem, relationship of Lipschitz functions and W^{1,\infty}

2. Hausdorff measure and dimension

  • outer Hausdorff measure
  • Hausdorff measure
  • Hausdorff dimension
  • connections to Lebesgue measure
  • area formula (without a proof)

Last update: Zelený Miroslav, doc. RNDr., Ph.D. (21.09.2022)
Entry requirements -

Differential calculus of several variables, basic theory of metric spaces, theory of measure and Lebesgue integral (as covered by lecture NMMA203).

Last update: Zajíček Luděk, prof. RNDr., DrSc. (08.05.2018)