Functional Analysis/Funktionalanalysis

Lecturer: Roland Speicher

Assistant: Felix Leid

News

  • There will be an oral exam at the end of the term. To be admitted to the exam, you are expected to achieve at least half of the available points on the assignments.
  • Felix Leid can be reached by email under leid@math.uni-sb.de
  • There is now also a mailbox (009) available for handing in the assignments.
  • The tutorials (Übungen) will be held on Fridays, 12 -14. The first tutorial will take place on November 11.
  • There is no lecture on Monday, 24. October. The course starts with the first lecture on Wednesday, 26. October.

Time and Place

The lectures and tutorials will be held in person.

Lectures (given by Roland Speicher)

Mondays 10-12 and Wednesdays 10-12 in HS IV, E 2.4

We start at 10 c.t. (i.e., 10:15).

Tutorials (given by Maximilian Leist)

Fridays 12-14, room: SR 9, E2.4

Office Hours

Roland Speicher: by appointment

Felix Leid: Tuesdays 14-15, room 219 in E 2.4; email: leid@math.uni-sb.de

Maximilian Leist: Mondays 13 - 14, Gruppenarbeitsraum in E 2.4

Assignments

The assignments can be handed in in class on Mondays or be thrown in mailbox 009 before the class on Monday.

Assignment 1 (due Nov 7)

Assignment 2 (due Nov 14) (note that in Problem 1b the span was missing in the first version of the assignment; and in the definition of l2 in Problem 3 the square is still missing)

Assignment 3 (due Nov 21)

Assignment 4  (due Nov 28)

Assignment 5 (due Dec 5). Note: there is a typo in 2a); in the definition of the set Fn, f(h) should be f(x).

Scans of Lecture Notes

By Lectures By Topics

General Information

  • This course is a Core Course (Stammvorlesung) in pure mathematics, worth 9 credit points.
  • The language of the course is English by default, unless all participants speak German.
  • We will roughly follow the course notes Functional Analysis by Moritz Weber.
  • To get an idea about some of the topics that we will cover in this course, you might watch the videos on Hilbert spaces and bounded operators on Hilbert spaces from a lecture series on the mathematics of quantum mechanics by the lecturer a few years ago. However, in the Functional Analysis course we will cover more topics on bounded operators, in particular in regard of their spectral theory.

Prerequisites

The course "Functional Analysis" builds on basic knowledge in analysis and linear algebra, at least on the level of our Analysis 1, Analysis 2, Lineare Algebra 1 classes. Having taken Analysis 3 and/or Funktionentheorie (Complex Analysis) might be helpful at some places, but is not a prerequisite for the class.

A condensed version of measure and integration theory, which might be helpful for this class, was given by the lecturer in a few videos as an appendix to a class on the mathematics of quantum mechanics, a few years ago.

Content

Functional Analysis deals with (infinite-dimensional) vector spaces, their topologies and geometries and in particular linear mappings on those spaces. Roughly, one is trying to solve linear equations in infinite dimensions. Methods of linear algebra are thus extended to infinite dimensions, but have to be combined with analytic and topological considerations, and quite some new (and interesting) phenomena occur.

The methods of functional analysis are important in many fields, e.g., quantum mechanics, partial differential equations, differential geometry, non-commutative geometry.

We will provide in this class the basic functional analytic concepts and results, like Banach and Hilbert spaces, locally convex vector spaces, Banach algebras and C*-algebras, Theorem of Hahn-Banach, Theorem of Baire, representation theorem of Riez. An important point will be spectral theory, i.e., the generalization of eigenvalues and diagonalization of matrices to the infinite-dimensional setting.

Literature

  • John Conway, A course in Functional analysis, Springer, 1990
  • Reinhold Meise und Dietmar Vogt, Einführung in die Funktionalanalysis, vieweg, 1992
  • Gert Pedersen, Analysis Now, Springer, 1989
  • Orr Moshe Shalit, A First Course in Functional Analysis, Chapman and Hall/CRC 2020
  • Dirk Werner, Funktionalanalysis, Springer 2011

There is also a Semesterappart for the class in the library.

Postal address

Saarland University
Department of Mathematics
Postfach 15 11 50
66041 Saarbrücken
Germany

 

Visitors

Saarland University
Campus building E 2 4
66123 Saarbrücken
Germany