MAMO3300 Real analysis Course description

The course covers metric spaces, elementary topology, Hilbert spaces, self-adjoint and compact operators. Fundamental concepts in analysis such as convergence, continuity, completeness, and compactness are generalized to infinite-dimensional spaces where the elements are functions rather than vectors. The course has applications in quantum physics and provides the theoretical foundation for further studies in analysis.

The course builds on

• MAMO2300 - Linear algebra and introduction to group theory
• DAFE1000 - Mathematics 1000
• DAPE2000 - Mathematics 2000 with statistics

None.

After completing the course, the student is expected to have achieved the following learning outcomes defined in terms of knowledge, skills and general competence.

Knowledge

The student can:

• understand the distinction between real and rational numbers
• understand convergence and the delta-epsilon definition of continuity
• generalize convergence and continuity to metric and topological spaces
• identify examples of Banach spaces and Hilbert spaces
• understand different types of operators, such as compact and self-adjoint operators, and a spectral theorem for such operators.
• formulate quantum physical concepts within a mathematical framework.

Skills

The student can:

• determine convergence and continuity rigorously and generalize this to metric and topological spaces
• determin topological properties such as compactness for concrete examples of spaces
• determine whether relevant function spaces are Banach spaces
• perform concrete calculations with operators on Hilbert spaces

General competence

The student can:

• transfer analysis concepts from calculus to more general spaces that are of practical significance
• express themselves precisely within a rigorous mathematical language
• understand the mathematical apparatus behind quantum physics

Lectures and exercises: The main part will be lectures in plenary. In the exercise sessions, we work on tasks that are solved individually and in groups, and which are discussed or possibly presented. The aim is to engage the students throughout the semester.

None.

Individual written exam of 3 hours under supervision.

The exam result can be appealed.

In the event of resit or rescheduled exams, another exam form may also be used. If oral exams are used, the result cannot be appealed.

All written and printed aids are allowed. Handheld calculator that cannot be used for wireless communication or to perform symbolic calculations. If the calculator has the capability for internal memory storage, the memory must be cleared before the exam. Random checks may be carried out.

Grade scale A-F

One internal examiner. External examiners are used regularly.