Lineär funktionalanalys

**Valid from:** Autumn 2020**Decided by:** Professor Thomas Johansson**Date of establishment:** 2020-09-24

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** MATP35**Teaching language:** English

To give the postgraduate student a solid knowledge about the basic concepts of functional analysis, and about some of its applicatons. Such knowledge is important for research in mathematical analysis, and for research within some other fields of mathematical character such as control theory.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to give a detailed account of the concepts, theorems and methods within functional analysis that are treated in the course
- be able to identify the main theorems of the course, describe the main ideas in their proofs, and carry out the steps.
- be able to give examples of non-trivial situations where these theorems apply.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to integrate knowledge from the different parts of the course in connection with problem solving
- be able to identify problems that can be solved by methods that are part of the course and use appropriate solution methods
- be able to explain the solutions of related mathematical problems, in speech and in writing, in a logically coherent manner, and with adequate terminology.

*Judgement and Approach*

For a passing grade the doctoral student must be able to identify situations in which the methods of functional analysis apply, for example in other mathematical fields such as ordinary and partial differential equations, function spaces and operator theory.

The course treats fundamental properties of Banach and Hilbert spaces and the bounded linear operators defined on them: • Banach spaces, the Hahn-Banach Theorem, weak convergence and weak precompactness of the unit ball. • Hilbert spaces. Examples including L^2 spaces. Orthogonality, orthogonal complement, closed subspaces, the Projection Theorem. The Riesz Representation Theorem. • Orthonormal sets, Bessel's Inequality. Complete orthonormal sets, Parseval's Identity. • The Baire Category Theorem and its consequences for operators on Banach spaces (the Uniform Boundedness, Open Mapping, Inverse Mapping and Closed Graph theorems). Strong convergence of sequences of operators. *Bounded and compact linear operators on Banach spaces and their spectra. *The Spectral Theorem for compact self-adjoint operators on Hilbert spaces.

Lax, Peter D.: Functional Analysis. John Wiley & Sons, 2002. ISBN 9780471556046.

**Types of instruction:** Lectures, seminars

**Examination formats:** Written exam, oral exam**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** Mathematics correponding to the master programme Engineering Mathematics and some course on Lebesgue integration.

**Course coordinators:** **Web page:** http://www.ctr.maths.lu.se/course/MATP35/