# Exam-Summer Semester

The date of the summer exam is 23rd July 2019 (8:00-10:00) SR 2.067 (Math building 20.30)

# Exam

The results are posted to a notice board near the office 2.027 in 20.30 Math building.

# Klausureinsicht

Exams can be viewed on Thursday 2nd May from 13:00 to 14:00 in 20.30 Math building (SR 1.067).

In this lecture we study Banach and Hilbert spaces and linear operators between them. The focus is on infinite-dimensional spaces and examples include function and sequence spaces. Linear operators on such spaces arise in the formulation and solution of integral and differential equations, and the development of Functional Analysis in the 20th century has been intimately linked to the modern theory of such equations.

Functional Analysis as a "common language" is the basis for advanced studies in a number of fields such as partial differential equations, numerical analysis, mathematical physics, and a lot more.

The topics we shall study in this lecture include

- metric spaces (notions of topology, compactness)
- continuous linear operators on Banach spaces
- uniform boundedness principle and open mapping theorem
- Hilbert spaces, orthonormal bases, Sobolev spaces
- Dual spaces, Hahn-Banach and Banach-Alaoglu theorems, weak convergence, reflexivity
- compact linear operators.

Prerequisites: Analysis I-III, Linear Algebra I-II

# Summary of the Lecture

The summary (version 06.02.19) will be constantly updated.

# Exercise sheets

# Examination

Written exam is taking place at 20th March 2019, 11:00-13:00

in 20.40 Fritz-Haller Hörsaal (HS37).

# References

H. Brezis: Functional Analysis, Sobolev Spaces, and Partial Differential Equations.

J.B. Conway: A Course in Functional Analysis.

M. Haase: Functional Analysis: An Elementary Introduction.

M. Reed, B. Simon: Functional Analysis.

W. Rudin: Functional Analysis.

R. Meise, D. Vogt: Introduction to Functional Analysis.

There is also a list of books in German that should be mentioned:

H.W. Alt: Lineare Funktionalanalysis: Eine anwendungsorientierte Einführung, 4. Auflage, Springer 2012.

M. Dobrowolski: Angewandte Funktionalanalysis, Springer 2006.

H. Heuser: Funktionalanalysis: Theorie und Anwendung, 4. Auflage, Teubner 2006.

D. Werner: Funktionalanalysis, 8. Auflage, Springer 2018.

J. Wloka: Funktionalanalysis und Anwendungen, de Gruyter 1971.