Department Mathematik
print


Navigationspfad


Inhaltsbereich

Exercises in
Functional Analysis, summer 2016

This site provides material for the exercise and tutorial classes complementing the lecture course Functional Analysis held by Prof. Thomas Ø. Sørensen, PhD.

News

21.07.2016 This site will no longer be updated (check here for news about the re-exam).
18.07.2016 The results of the exam are now ready. They can be seen in the glass box ('Schaukasten') in front of B 409 (Siedentop). You may have a look at your exam ('Klausureinsicht') on Wednesday July 20th, at 13:00, in B 006.
15.07.2016 In the exam, you may use results from the lecture and from the homework sheets (without re-proving them) IF you tell what you use ('from lecture' is not enough) - except of course when explicitely asked to prove something.
13.07.2016 Exercise Sheet 12 will be marked until Friday morning. You can expect to find your sheets in the return box on the first floor before noon.
06.07.2016 Note that in Problem 4 on Exercise Sheet 11, for the original statement of the problem, it was not necessary to use the Closed Graph Theorem. However, by using the Closed Graph Theorem, it can be shown that, for the operator T to be bounded, it is not necessary to assume that the map J has closed range (see solution below). We have changed the statement of the problem accordingly. Clearly, this change does not affect the grading of your solutions.
29.06.2016 Information about the exam is online.
22.06.2016 As noted on several occasions, in the literature, the term pre-compactness is used inconsistently. Sets that in the lecture are called pre-compact are sometimes referred to as totally bounded, while pre-compact is then used interchangeably with relatively compact. In Assignment 10, Problem 2, we want you to prove whether the sets are relatively compact (not pre-compact as falsely written on the first version of the sheet).
11.06.2016 Three minor errors on the current exercise sheet (8) have been corrected (2c(i): x non-zero, 3b: a non-zero, 4b: for all functionals f on c_0).
06.06.2016 The re-exam (Wiederholungsklausur) will take place Wednesday, July 27th 2016 (27.07.2016), from 9:15 to 11:45.
20.05.2016 Due to the holiday next Thursday, we encourage the students attending Thursdays tutorial class to go to one of the other classes next week. Marked sheets that would have been returned in the tutorial can be found in the return box next to the library.
10.05.2016 Due to the holiday on next Monday and the lecture-free Tuesday, we encourage the students attending the tutorial class on Monday to attend one of the other classes this week and the students attending the tutorial class on Tuesday to go to one of the other classes next week (preferably not to the class on Thursday, which has 20+ students already). Marked sheets that would have been returned in one of the cancelled tutorials can be found in the return box next to the library.
10.05.2016 Due to the holiday on next Monday and the lecture-free Tuesday, we encourage the students attending the tutorial class on Monday to attend one of the other classes this week and the students attending the tutorial class on Tuesday to go to one of the other classes next week (preferably not to the class on Thursday, which has 20+ students already). Marked sheets that would have been returned in one of the cancelled tutorials can be found in the return box next to the library.
03.05.2016 Due to the holiday on Thursday, we encourage the students attending Thursday's tutorial class to go to one of the other classes this week.
03.05.2016 The phrasing of Exercise 1c on Exercise Sheet 3 was slightly changed.
29.04.2016 An error on Exercise Sheet 2 was corrected (Problem 2d: ...is the basis for a topology...).
27.04.2016 A typo and an error on Exercise Sheet 2 was corrected (Problem 1d: super set instead of subset, Problem 2c: B is the basis for a topology, not necessarily T).
19.04.2016 Starting April 21, the lecture on Thursday will be 8.25-9.55 (and not 8.30-10.00).
12.04.2016 The registration is now enabled.
07.04.2016 Here are important dates for the first two weeks:
First lecture: Tue, April 12 2016
First homework discussion: Mo, April 18 2016
First tutorial class: Tue, April 19 2016
Activation of the registration: Tue, April 12 2016 (evening)
Deadline for the hand in of Assignment 1: Mo, April 18 2016, 14:00

Registration

Please register now.

In the process of registration for the tutorial classes you may choose one out of five dates (see the timetable below). Take your time in choosing the correct group, since you won't be able to change it afterwards by yourself. If you just want to do the weekly homework assignments without participating in the tutorial sessions, then please choose group X.

The data submitted by using the registration form will be kept private and won't be published at any time.

Timetable

Time
Monday Tuesday Wednesday Thursday Friday
8
to
10
Lecture
Prof. Sørensen
C 123
10
to
12
12
to
14
Lecture
Prof. Sørensen
C 123
Tutorial C
B 045
14
to
16
Tutorial E
B 251
Tutorial A
B 045
Tutorial D
B 039
16
to
18
Homework
Discussion

Andreas Groh
C 123
Tutorial B
B 045
Remark. Even though the lecture is held in English, you are free to speak German in the tutorial classes. Tutorial D will be in English.

Exercise sheets

Date
Homework
Tutorial
Main topics
Due
Mo, July 11
--
Tutorial 13
Weak and weak* convergence, reflexive spaces.
--
Mo, July 4
Assignment 12
Tutorial 12
Hahn-Banach Theorem,
weak convergence.
Mo, July 11, 14:00
Mo, June 27
Assignment 11
Tutorial 11
Baire, Open and Inverse Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle.
Mo, July 4, 14:00
Mo, June 20
Assignment 10
Tutorial 10
Compact operators, orthonormal
bases.
Mo, June 27, 14:00
Mo, June 13
Assignment 9
Tutorial 9
Inner product spaces, Minkowski inequality, duals of sequence spaces.
Mo, June 20, 14:00
Mo, June 6
Assignment 8
Tutorial 8
Bounded linear functionals and their norms, dual spaces.
Mo, June 13, 14:00
Mo, May 30
Assignment 7
Tutorial 7
Separability of sequence spaces, bounded linear maps, operator norms.
Mo, June 6, 14:00
Mo, May 23
Assignment 6
Tutorial 6
Banach spaces, sequence spaces, absolute convergence
Mo, May 30, 14:00
Mo, May 16
Assignment 5
Tutorial 5
Compactness, contractions, and examples of function spaces
Mo, May 23, 14:00
Mo, May 9
Assignment 4
Tutorial 4
Metric spaces, compactness, and homeomorphisms
Wed, May 18, 10:00
Mo, May 2
Assignment 3
Tutorial 3
Separability, 1st and 2nd countability, product topology, initial topology
Mo, May 9, 14:00
Mo, Apr 25
Assignment 2
Tutorial 2
Topological basics II (Closure, interior, boundary, base, continuity)
Mo, May 2, 14:00
Mo, Apr 18
Assignment 1
Tutorial 1
Topological basics I (Closure, interior, boundary, relative topology)
Mo, Apr 25, 14:00
Mo, Apr 11
Assignment 0
--
Simple exercises from earlier analysis courses (to train writing maths)
Mo, Apr 18, 14:00

Solutions to exercise sheets not discussed in class: 2, 3, 4, 5, 7, 8, 9, tut 9, 10, 11.

Team

Lecturer. The lectures are held by Prof. Østergaard Sørensen, PhD.

Assistants. The exercise and tutorial classes are organized by Andreas Groh and Sebastian Gottwald.

Correctors. Don't hesitate to ask questions during the tutorial classes even if the questions may not be directly related to the current exercise sheet. If you have a question regarding the grading of your exercise sheets, you can find the mail addresses of the correctors below.

Token Marker E-Mail
CK Christoph Kehle
FR Francesco Romano
JL Jingjia LIU
KR Kilian Rückschloß