Functional Analysis

Functional Analysis

Dates and times

Lecture

Tuesday, 14:15-15:45, HS 4, Building E2 4
Thursday, 14:15-15:45, HS 4, Building E2 4

Tutorial

Thursday, 8:30 - 10:00, HS 4, Building E2 4

Contents

Functional analysis provides an abstract framework for the study of many questions in analysis. A basic idea behind functional analysis is to regard the objects of interest, for example sequences or functions, as points in a vector space. To obtain a meaningful theory, one endows the vector space with a suitable norm, which makes it possible to talk about concepts such as convergence or continuity. Thus, functional analysis blends together ideas from linear algebra and from analysis.

This course will provide an introduction to the fundamental concepts of functional analysis. In particular, this includes the following topics:

• Normed spaces
• Linear operators
• Dual spaces and Hahn-Banach theorem
• Open mapping theorem and uniform boundedness principle
• Hilbert spaces
• Banach and C*-algebras
• Spectral theory

Language

The course will be taught in English unless all participants speak German.

Prerequisites

Analysis 1-3 and Funktionentheorie (Complex Analysis), as taught at UdS, are sufficient.

Detailed prerequisites:

Students are expected to be proficient in the following topics, including proof techniques:

• Real analysis of metric spaces: Open and closed sets, convergence of sequences, completeness, compactness, continuity of functions, uniform continuity, uniform convergence
• Linear algebra: Abstract vector spaces, dual space, quotient spaces, linear mappings, eigenvalues and eigenvectors

Familiarity with the following topics is recommended:

• Measure theory: Lebesgue integration in measure spaces, monotone and dominated convergence theorem, Fubini's theorem
• Complex analysis: Holomorphic functions, Cauchy integral formula, Liouvilles theorem, Laurent series

If you need reading recommendations to review some of these topics, please talk to the instructor.

Assignments

There will be weekly assignments. Submission is via CMS. You can submit assignments in teams of up to three people.

Exam

There will be an exam at the end of the semester, most likely oral. In order to be eligible for the exam, you have to achieve at least 50% of the available points on the assignments.

Literature

See here.