Syllabus for Foundations of Mathematical Analysis

Learning outcomes

The purpose of the course is to give basic knowledge in real analysis and functional analysis to students who has not already taken these courses.

On the completion of the course the student should be able to:

• describe the construction of the real line and its properties,
• describe the principles behind differential and integral calculus, which includes being able to give definitions as well as proofs of the important theorems from the real analysis,
• describe the properties of the linear operators in Banach och Hilbert spaces,
• state the hypothesis and explain the proof of the Spectral theorem for compact operators,
• apply the above-mentioned theory in problem solving as well as simple proofs.

Content

Real analysis: Definitions and properties of real numbers. Cauchy sequences, open and closed sets, compact sets, Heine-Borel lemma. Continuous functions. Differentiable functions: mean value theorem with corollaries. Taylor series. Stone-Weierstrass theorem. Banach fixed point theorem.

Functional analysis: Banach spaces. Fundamental theorems in functional analysis. Linear operators on Banach spaces. Hilbert spaces and operators on them. The spectral theorem for compact operators.

Instruction

Lectures.

Assessment

Assignments during the course and oral examination.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Other directives

This course cannot be included in the same degree as 1MA331, 1MA226 or 1MA218.