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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

• 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

• 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

• 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

• home assignments
• midterm
• final exam

#### Interim Assessment

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

#### Recommended Core Bibliography

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