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Syllabus:

# Real Analysis, 7.5 Credits

Swedish name: Reell analys

This syllabus is valid: 2024-08-19 and until further notice

Course code: 5MA182

Credit points: 7.5

Education level: First cycle

Main Field of Study and progress level: Mathematics: First cycle, has at least 60 credits in first-cycle course/s as entry requirements

Established by: Faculty Board of Science and Technology, 2017-06-12

Revised by: Faculty Board of Science and Technology, 2023-12-15

## Contents

The course provides advanced knowledge of concepts and theorems in advanced analysis. The concept of topology is introduced in
metric spaces. The concepts of compactness and continuity are essential. Thereafter real-valued functions defined on
metric spaces are studied, with a focus on continuity and function sequences. Central theorems are Heine-Borel covering theorem, Urysohn's lemma and Weierstrass' approximation theorem. The concept of differentiability of vector-valued functions is introduced and
the inverse and implicit function theorems are proved.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• describe in detail definitions and theorems concerning metric spaces and real-valued function theory
• describe in detail the notions of pointwise and uniform convergence

Skills

• construct metric spaces and functions with given regularity
• conduct formal proofs concerning metric spaces and real-valued functions
• determine sets' topological properties
• determine real-valued function regularity properties

## Required Knowledge

The course requires 60 ECTS in mathematics or at least two years university studies. In both cases requires 15 ECTS in Calculus and 7,5 ECTS in Linear Algebra or equivalent. Proficiency in English equivalent to the level required for basic eligibility for higher studies. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

## Form of instruction

Teaching is mainly in the form of lectures.

## Examination modes

The course is examined by a written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).
A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

## Other regulations

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.

## Literature

### Valid from: 2024 week 34

Rudin Walter
Principles of mathematical analysis
3. ed. : New York : McGraw-Hill : cop. 1976 : 342 s. :
ISBN: 0-07-054235-X
Mandatory
Search the University Library catalogue