## Functional Analysis [MA3001]

### Wintersemester 2019/20

### Prof. Dr. Robert Koenig

Dozent: |
Prof. Dr. Robert Koenig | |

Übungsleitung: |
Margret Heinze | |

Mitwirkende: |
Markus Hasenoehrl, Alexander Kliesch, Frederik vom Ende | |

Vorlesung: |
Tuesday, 14:00 - 16:00, PH HS 2, and Friday, 14:00 - 16:00 Rudolf-Mößbauer-HS |
Anmeldung |

Tutorübungen: |
Mon 8:30-10:00, 02.08.011 (Hasenoehrl) Mon 12:15-13:45, 02.08.011 (Heinze) Mon 16:15- 17:45, 00.09.022 (Kliesch) Tue 8:30-10:00, 03.08.011 (vom Ende) Tue 10:15-11:45, BC2 0.01.04 in Hochbrueck (Heinze) |
Anmeldung |

### News

- Updates on the second (Nachprüfung) exam, cf. below.
- The second exam (Nachprüfung) has been cancelled and will be rescheduled because of the current Coronavirus situation.
- The first exam has been evaluated and results are posted. An opportunity for viewing one's exam ("Einsicht") will be provided at a later point.
- Final version of the exam cheat sheet (containing theorems from the lecture notes) has been uploaded (will be attached to the exam)
- Open question tutorials take place until the exam, i.e., Wednesday 19th and 26th from 16:00-18:00 in room 03.06.011
- Exercise sessions on Jan 20 and 21: review of the mock exam. There will be no tutorial exercises and no homework to submit in that week.
- On Friday Jan 17: Mock exam during lecture time (14:00 - 16:00, PH HS 2): Theorems from the lecture as Mock exam cheat sheet for the mock exam (will be provided for you).
- Information on exercise sessions on 23.12.: Sessions 12:15-13:45 and 16:15-17:45 will take place regularly, session 8:30-10:00 is cancelled.
- The exercise class 4 on Tuesday November 12, 10:15-11:45 will be cancelled due to the Fachschaftsvollversammlung: Please find another exercise class that you can attend during this week.
- The lecture on Friday, November 8 will take place in Garching-Hochbrueck BC2 0.01.17, Hörsaal (8102.EG.117) starting at 2.15pm. map
- On Wednesday November 6, the special/question exercise session (16:00-18:00 in room 03.06.011) is cancelled.
- Exercise sessions will be held every week starting from October 21.
- Please sign up for exercise sessions in TUMOnline before Sunday, October 20.

### Content (from Module description)

Banach and Hilbert spaces; bounded linear operators, open mapping theorem; spectral theory for compact selfadjoint operators; duality, Hahn-Banach theorems; weak and weak* convergence; brief introduction to unbounded operators### Second Exam (Nachprüfung)

- The exam (Nachholklausur) is scheduled for the 24th August 2020 (13:00-14:30).
- When signing up for this exam, you are given a choice of

- ) writing the exam at home/a location of your choosing under virtual video supervision or
- ) writing the exam under supervision at the department of mathematics.

- To familiarize yourself with the new format 1), we will offer a short “mock exam” (two possibilities on Thursday, August 13, 5.30pm or on Monday, August 17, 1pm. If you signed up for the exam, you should receive more details via e-mail).
- We note that it is up to you to decide which of the options 1) or 2) to choose. (Our hope is that the majority of you will choose option 1). Independently of your choice, we will strive to minimize social contacts/maximize interpersonal distance to make sure that taking part in this exam does not involve unnecessary health hazards.)

### Exam

- The exam will take place on
**Friday**, 28.02.2020 at 8:30 in MW 0001, Gustav-Niemann-Hörsaal (5510.EG.001) Duration: 90min -
~~The second exam (Nachprüfung) will take place on~~**Tuesday**, 14.04.2020 at 15:00 in 003 HS 2, "Interims II" (5416.01.003) Duration: 90min - You are NOT allowed to bring any handwritten notes to the exam. Instead, we will attach the following cheat sheet (containing all the theorems) to the exam.

### Exercises

- There will be two types of problems: Tutorial problems should be worked on and discussed during the exercise sessions. The remaining problems are assigned as homework.
- In addition to the regular exercise sessions, there will be a special exercise session on Wednesdays (16:00-18:00 in room 03.06.011). This will be a session for students who have additional questions or need help. Students can come to solve their exercises there and ask questions concerning concepts from the lecture.
- New exercise sheets are typically posted online on Fridays every a week.

### Grading/Bonus system

- Collaboration in groups of (up to) three students possible.
- Solutions are to be submitted into a letter box (number 04, in the basement of the MI building) until 14:00 on the day indicated on the respective exercise sheet (usually Fridays)
- Solutions will be marked in a binary fashion with grades 'adequate' and 'inadequate'
- On the homework sheets please indicate the Matrikelnummer of all (up to three) students. Also provide the name of a TA responsible for an exercise group one of you belongs to.
- Solutions are returned to you during the exercise sessions in subsequent weeks.
- A bonus will be awarded if the grade 'adequate' is earned on at least 75% of the exercises: This means that you have to receive an 'adequate' on at least 153 of the 204 exercise (parts).
- Bonus system: A bonus means that the grade obtained in the final exam is upgraded by one step on the grading scale. However, only those students whose grade is sufficient for passing the exam can benefit from this bonus. Bonus credits cannot be transferred to functional analysis courses/exams held in the future.

Exercise Sheet | Week | Date for homework submission | Solution |
---|---|---|---|

Ex01 | Oct. 21 - Oct. 25 | Oct. 25 | Sol01 |

Ex02 | Oct. 28 - Nov. 1 | Nov. 1 | Sol02 |

Ex03 | Nov. 4 - Nov. 8 | Nov. 8 | Sol03 |

Ex04 | Nov. 11 - Nov. 15 | Nov. 15 | Sol04 |

Ex05 | Nov. 18 - Nov. 22 | Nov. 22 | Sol05 |

Ex06 | Nov. 25 - Nov. 29 | Nov. 29 | Sol06 |

Ex07 | Dec. 2 - Dec. 6 | Dec. 6 | Sol07 |

Ex08 | Dec. 9 - Dec. 13 | Dec. 13 | Sol08 |

Ex09 | Dec. 16 - Dec. 20 | Dec. 20 | Sol09 |

Ex10 | Dec. 23 - Jan. 10 | Jan. 10 | Sol10 |

Ex11 | Jan. 13 - Jan. 17 | Jan. 17 | Sol11 |

Mock exam | no submission | Mock Sol | |

Ex12 | Jan. 27 - Jan. 31 | Jan. 31 | Sol12 |

Ex13 | Feb. 3 - Feb. 7 | Feb. 7 | Sol13 |

### Literature

There are many good books and lecture notes. Here are some of them. Apart from Bollobas and Brokate they go considerably beyond what we can do in one semester.- Walter Rudin: Functional Analysis (McGraw Hill, 1991)
- M Reed, B Simon: Functional Analysis (Academic Press, 1972)
- D Werner: Funktionalanalysis (Springer, 2007)
- F Hirzebruch, W Scharlau: Einfuehrung in die Funktionalanalysis (BI-Hochschulbuecher, 1991)
- Peter D. Lax: Functional Analysis (Wiley, 2002) [good two semester course]
- Gert K. Pedersen: Analysis Now (Springer, 1989) [very compact, very elegant, but quite advanced]
- John B. Conway: A Course in Functional Analysis
^{}(Springer, 1990) [standard book on the subject; good two semester course] - Dirk Werner: Funktionalanalysis (Springer, 1995) [good two semester course; German]
- Bela Bollobas: Linear Analysis
^{}(Cambridge University Press, 1990) [well written; based on a course given in Cambridge; closest to our lecture course] - Martin Brokate: Funktionalanalysis (Vorlesungsskript) [good German script based on the course taught in WS13/14 at TUM] and Functional Analysis (english version)

### Lecture Notes

Lecture notes will be provided here: current version (updated on Feb 18: last sentence of Def. 1.4.6. (product topology) corrected). Any corrections/suggestions are welcome. * Clarification about Remark 7.1.15.Additional materials/slides | Date updated |
---|---|

Metric and Topological spaces | Oct 18 |

Posets and Nets | Nov 5 |

Geometric Hahn-Banach Theorems | Dec 6 |