Course syllabus - Foundations of Real Analysis 7.5 credits
|Valid from:||Autumn semester13 Autumn semester19|
|Level of education:||Second cycle|
|Main Field(s) of Study:||Mathematics/Applied Mathematics,|
|In-Depth Level:||A1N (Second cycle, has only first-cycle course/s as entry requirements),|
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.
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.
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.
Lectures and classes.
Specific entry requirements
Calculus of Several Variables 7.5hp or corresponding.
INL1, 2.5 credits, marks Pass (G), Assigned problems
TEN1, 5.0 credits, marks 3, 4 or 5, Written examination
A student who has a certificate from MDH 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 (2016/0601). 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.
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Course literature is not yet public.