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Regular version of the site
2020/2021

Research Seminar "Geometric Foundations of Analysis"

Type: Optional course (faculty)
When: 1, 2 module
Language: English
ECTS credits: 6

Course Syllabus

Abstract

We introduce and study the basic notions of analysis: metric spaces, continuity, differentiation and integration. We also prove some applications, such as the mean value theorem, and Cauchy inequality.
Learning Objectives

Learning Objectives

  • The goal of the course is to teach the students to handle the basic notions of analysis: metric spaces, continuity, differentiation and integration.
Expected Learning Outcomes

Expected Learning Outcomes

  • The students learn the notion of metric spaces
  • The students learn the notion of continuous maps
  • The students learn the notion of complete metric spaces
  • The students learn the notion of compactness
  • The students learn the notion of absolutely convergent series
  • The students learn the notion of differentiation
  • The students learn the mean value theorem and its applications
  • The students learn the notion of integration
Course Contents

Course Contents

  • Metric spaces
    We define metric spaces and study their basic properties
  • Continuous maps
    We define continuous maps and study their basic properties
  • Integration
    We define integration and study its basic properties
  • The mean value theorem
    We prove the mean value theorem and discuss its applications.
  • Derivatives
    We define derivatives of functions of one real variable and study their basic properties.
  • Absolutely convergent series
    We define absolutely convergent series and study their properties
  • Compactness
    We define compact metric spaces and study their basic properties
  • Complete metric spaces
    We define complete metric spaces and study their basic properties
Assessment Elements

Assessment Elements

  • non-blocking home assignments
  • non-blocking midterm
  • non-blocking final exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.5 * home assignments + 0.5 * midterm
Bibliography

Bibliography

Recommended Core Bibliography

  • Rudin, W. (1976). Principles of mathematical analysis.

Recommended Additional Bibliography

  • V. A. Zorich. (2016). Mathematical Analysis I (Vol. 2nd ed. 2015). Springer.