# Functional Analysis/Funktionalanalysis

### Lecturer: Roland Speicher

### Assistant: Felix Leid

## News

**Oral Exams:**For getting credit for the course one has to pass an oral exam. Possible slots for the exams are in the periods**March 6 - 9**or**March 30 - 31**. In order to arrange a date for the exam (or for other questions in this context) you should contact Felix Leid.- There will be no lecture on Monday, February 6.Instead, read the material in the notes and watch the following videos from the Math of QM lecture series:
- Motivation of spectral theorem via finite-dimensional situation and the multiplication operator in infinite dimensions
- Motivation of "resolution of identity" (=spectral measure) and spectral theorem via multiplication operator
- and maybe some more
- The next (and last) lecture will be on Wednesday, Februar 8.
- There is now also a set of videos on the Stone-Weierstrass Theorem.
- As announced there will be some videos on background material for the class; the first such set, on the Theorem of Arzela-Ascoli, is recorded now.
- The tutorial in the week of Dec 19 will be held online, via Microsoft teams; you can request to be added to the team via this link
- For questions on the online tutorial or how to hand in the assignment on Monday, Dec 19 please contact Maximilian Leist via email: male00005@stud.uni-saarland.de
- In the two weeks before and after the Christmas Break (i.e., the weeks of Dec 19 and of Jan 2) the university buildings will be closed and teaching will be moved to online. This means in our case:
- there will be no lectures during those two weeks, but we will upload a few videos on specific topics - more details on this later;
- there will be an online tutorial, to discuss assignment 7, in the week of Dec 19, but no tutorial in the week of Jan 2; assignment 8 will be uploaded sometimes during the break and will be due on Jan 9.
- 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.

- Lebesgue-Stieltjes integration, part 1
- Lebesgue-Stieltjes integration, part 2 (including the definition of measurable and integrable functions)
- Convergence Theorems
- Sets of Measure Zero
- Absolutely Continuous Functions, part 1
- Absolutely Continuous Functions, part 2

## 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 l_{2} 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 F_{n, }f(h) should be f(x).

Assignment 6 (due Dec 12)

Assignment 7 (due Dec 19) [note that the university buildings will be closed in the week of Dec 19; there will be no lectures, but the tutorial will take place during this week - you should arrange with Maximilian Leist how to hand in the assignment and how the tutorial will be arranged]

Assignment 8 (due January 9)

Assignment 9 (due January 16)

Assignment 10 (due January 23) (Note that in the first version of the assignment in Exercise 2 there was a typo in the definition of A/J; a ϵ J should be a ϵ A. This has been corrected now.)

Assignment 11 (due January 30)

Assignment 12 (due February 6)

## Scans of Lecture Notes

Lecture 1:

Lecture 2:

Lecture 3:

Lecture 4:

Lecture 5:

Lecture 6:

Lecture 7:

Lecture 8 (21.11.22):

Lecture 9 (23.11.22):

Lecture 10 (28.11.22):

Lecture 11 (30.11.22):

Lecture 12 (5.12.22):

Lecture 13 (7.12.22)

Lecture 14 (12.12.22)

Lecture 15 (14.12.22)

Lecture 16 (9.1.23)

Lecture 17 (11.1.23)

Lecture 18 (16.1.23)

Lecture 19 (18.1.23)

Lecture 20 (25.1.23)

Lecture 21 (30.1.23)

Lecture 22 (1.2.23)

Lecture 23 (6.2.23)

Lecture 24 (8.2.23)

- Introduction (Lecture 1)
- Hilbert Spaces (Lectures 1, 2, 3, 4, 5; most of this material is also covered in this youtube video)
- Bounded Operators on Hilbert Spaces (Lectures 6, 7; most of this material is also covered in this youtube video)
- Banach Spaces (Lectures 7, 8)
- The Hahn-Banach Theorem (Lectures 8, 9)
- Baire Category Theorem and Consequences (Lectures 10, 11)
- Geometry of Banach Spaces (Lecture 11)
- Adjoints of Operators on Banach Spaces (Lecture 12)
- Compact Operators on Banach Spaces (Lectures 12, 13)
- Spectral Theory of Compact Operators (Lectures 13, 14, 15)
- Banach Algebras (Lectures 16,17, 18, 19)
- Commutative Banach Algebras (Lectures 19, 20)
- The Gelfand Transform (including measurable functional calculus) (Lectures 21, 22)
- Spectral Theorem for Normal Operators on Hilbert Spaces (Lectures 23,24)

## 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