Elements of Functional Analysis

Recent news can be found on the German version of this website.

• Lectures: Monday, 10:15 - 12:00, Helmholtzstraße 18, room 220
• Exercise course: Wednesday, 12:15 - 13:00, Helmholtzstraße 18, room 220

During the first years of your studies you have already dealt with many mathematical equations whose solutions were mostly numbers or tuples of numbers.

Yet, in many applications equations occur whose solutions are not numbers, but functions or sequences of numbers. Examples of such equations are differential and integral equations which play an important role in finance, in science and engineering, and in stochastics.

In order to develop a strategy for solving those equations one has to understand the structure of all possible solutions, first. For example, the following questions are important:

• How can we compute the distance of two functions?
• What does it mean to say that a sequence of functions converges to another function?
• How can we define mappings which map functions to other function and why are these mappings important?
• What does it mean that two functions are perpendicular?
• Given an equation whose solution is supposed to be a function, how can we find out whether the equation is really solvable, and how can we compute the solution?

Functional analysis deals with these and with similar questions. For example, the notion of a "normed vector space" is introduced to answer the first two questions. The third question motivates the concept of a "linear operator" which occurs in many applications. The fourth question can be answered in the setting of so called "Hilbert spaces", a concept whose finite dimensional counter part you have already encountered in linear algebra. To answer the fifth question we give an introduction to "spectral theory". This is a very powerful tool to solve partial differential equations and integral equations; moreover, it is a fundamental concept in physics, in particular in quantum mechanics.

The contents of this course provide a foundation for many more advanced topics in mathematics such as

• analysis of partial differential equations
• numerics of partial differential equations
• Fourier analysis and its generalisations (e.g. wavelets)
• probability theory (e.g. stochastic processes and stochastic differential equations)
• mathematical foundations of quantum mechanics.

English literature:

• Peter D. Lax, Functional Analysis, Wiley, 2002
• Markus Haase, Functional Analysis: an elementary introdcution, American Mathematical Society, 2014

German literature:

• Harro Heuser, Funktionalanalysis: Theorie und Anwendungen, Teubner, 1992
• Dirk Werner, Funktionalanalysis, Springer, 2005
• Joachim Weidmann, Lineare Operatoren in Hilberträumen, Teil 1: Grundlagen, Teubner, 2000
• Wolfgang Arendt, Karsten Urban, Partielle Differentialgleichungen, Spektrum Akademischer Verlag, 2010
• Winfried Kaballo, Grundkurs Funktionalanalysis, Spektrum Akademischer Verlag, 2011

You can find the exercise sheets on the German version of this page.

To properly understand the contents of this course you need the following prerequisites:

• Analysis 1 and 2, Linear Algebra 1 and 2 (or similar courses such as Mathematics for Physicists 1 - 3)
• Measure Theory

The course is directed mainly to students of one of the following study programmes:

• Mathematics (Bachelor)
• Mathematics and Management (Bachelor)
• Mathematical Biometry (Bachelor)

Nevertheless, the course could also be of interest for students of other subjects, such as e.g. physics. If you are a student of another programme than the ones listed above and intend to take an exam in this course, please refer to the examination regulations of your study programme or contact the examination board to ensure that this is possible.

At the end of the course here will be an oral exam.