Course syllabus - Foundations of Real Analysis

Objectives

The course Foundations of Real Analysis aims at consolidating and deepening the students' knowledge of mathematical analysis acquired in elementary courses, and to prepare students for higher studies in mathematics, physics and technology.

Learning outcomes

At the end of the course the student is expected to be able to …
1. explain the basic concepts used in describing metric spaces topologically.
2. conclude whether sequences in metric spaces are convergent or not.
3. apply the concept of continuity for mappings between metric spaces.
4. apply the concept of differentiation for real functions. Special attention is paid to Taylor's theorem and special cases of it.
5. conclude for which functions the Riemann-Stieltjes integral exists.
6. conclude whether sequences of functions and series of functions are uniformly convergent or not, and to be able to apply this with respect to continuity, differentiability and integrability.
7. with the precise definitions of fundamental concepts occurring in mathematical analysis, in a logical correct way carry out and explain reasoning and proofs.

Course content

The real number system. The concept of convergence in metric spaces. The epsilon-delta definition of a limit, proofs of limit theorems. Basic topology: Countable, uncountable, compact, perfect, and connected sets. Numerical sequences and series: Convergence, upper and lower limits, convergence (criteria) of series, power series, absolute convergence, rearrangements. Continuity, uniform continuity, continuity and compactness, continuity and connectedness, discontinuities, monotonic functions. The derivative of a real function, mean value theorem, Taylor's theorem. The Riemann-Stieltjes integral, the fundamental theorem of calculus. Sequences and series of functions. Uniform convergence.

Tuition

Lectures and classes.

Specific requirements

At least totally 120 credits in the engineering, natural sciences, business administration or economics areas including Calculus of Several Variables, 7.5 credits, out of which 3.5 credits must be completed at the beginning of the course, or equivalent.
In addition Swedish course 3/Swedish course B and English course 6/English course A are required. For courses given entirely in English exemption is made from the requirement in Swedish course 3/Swedish course B.

Examination

INL1, Written assignment, 2.5 credits, written assignment concerning learning outcomes 1-7, grades Fail (U) or Pass (G).
HEM1, Take-home examination, 5 credits, examination concerning learning outcomes 1-7, grades Fail (U), 3, 4 or 5.

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Grading scale: 5, 4, 3