News
Last Problem Session (TEAMS Link below)Written on 13.02.24 (last change on 15.02.24) by Jonas Metzinger Dear Students, As I've already mentioned, the last Problem Session for Homework 14 will be this Friday starting at 2pm. This Problem Session will be online on Teams. The link for the Teams Meeting will follow… Read more Dear Students, As I've already mentioned, the last Problem Session for Homework 14 will be this Friday starting at 2pm. This Problem Session will be online on Teams. The link for the Teams Meeting will follow soon.
https://teams.microsoft.com/l/meetup-join/19%3ameeting_MzYwNDgyZTQtOWJmMi00NDA4LTg5YTktMjRkNTJkOThhYmY1%40thread.v2/0?context=%7b%22Tid%22%3a%2267610027-1ac3-49b6-8641-ccd83ce1b01f%22%2c%22Oid%22%3a%22313ca728-edfb-4a94-a9b6-9f1a57a69cbe%22%7d
Best Jonas |
Office HoursWritten on 07.02.24 by Alberto Dayan Dear students, As I mentioned in class, my office hours will continue until the exam, in case you have questions while you study. Apart from today's office hours, I will be available also on February 14th and February 21st, at 11 am in my office (room 2.18, building E2.4). If Wednesday doesn't… Read more Dear students, As I mentioned in class, my office hours will continue until the exam, in case you have questions while you study. Apart from today's office hours, I will be available also on February 14th and February 21st, at 11 am in my office (room 2.18, building E2.4). If Wednesday doesn't work for you, feel free to contact me via email, and we will schedule a meeting at another time during the week. Best, Alberto |
Typo in Problem 1, Homework 12Written on 23.01.24 by Alberto Dayan Dear students, there was a typo in Problem 1 of Homework 12, which has now been fixed. Best, Alberto |
Problem sessionsWritten on 23.01.24 by Jonas Metzinger Dear all, tomorrow we will talk about Homework 9 and 10. Next week will be about Homework 11 and 12. Note that you now should see how many points you still need to be admitted to the oral exam (50%). There are 14 Homeworks in total.
Best, Jonas |
Second Course in Functional AnalysisWritten on 13.01.24 (last change on 13.01.24) by Alberto Dayan Dear students, the Department asked me to gauge your interest in taking a second course in functional analysis during next semester. If you think that you will want to take a second course of functional analysis next semester, please let me know via email at dayan@math.uni-sb.de. The idea is that… Read more Dear students, the Department asked me to gauge your interest in taking a second course in functional analysis during next semester. If you think that you will want to take a second course of functional analysis next semester, please let me know via email at dayan@math.uni-sb.de. The idea is that the Department will try to offer such course, if enough students will show interest. Enjoy your weekend, see you in class next Tuesday! Best, Alberto |
Typo in Homework 8Written on 19.12.23 by Alberto Dayan Dear students,
problem 2 of Homework 8 contains a typo: T is assumed to be compact from X to Y, not just bounded. We saw many surjective operators taking values on infinite dimensional Banach spaces (for instance, take X=Y and T=id), so as it was written point (a) was false. I replaced the… Read more Dear students,
problem 2 of Homework 8 contains a typo: T is assumed to be compact from X to Y, not just bounded. We saw many surjective operators taking values on infinite dimensional Banach spaces (for instance, take X=Y and T=id), so as it was written point (a) was false. I replaced the pdf file with the correct one.
Best,
Alberto |
Calendar changes around Christmas and New YearWritten on 12.12.23 by Alberto Dayan Dear students,
as I mentioned in class earlier, there will be some change in the schedule in the weeks around Christmas and New Year. Here is a summary:
Dear students,
as I mentioned in class earlier, there will be some change in the schedule in the weeks around Christmas and New Year. Here is a summary:
Best,
Alberto |
Solutions for HomeworkWritten on 23.11.23 by Jonas Metzinger Dear students, Due to a lack of participation in the problem sessions, solutions for the homework will no longer be uploaded.
Best,
|
Change of Office HoursWritten on 02.11.23 (last change on 02.11.23) by Alberto Dayan Dear students,
due to a change in my schedule for the semester, I will move my office hours from Wednesdays, 16:00-17:00 to Wednesdays, 11:00-12:00. The location is the same, office 2.18 in E2.4. If none of the Wednesday time for the office hours and the Tuesdsay one (10:00 - 11:00) work… Read more Dear students,
due to a change in my schedule for the semester, I will move my office hours from Wednesdays, 16:00-17:00 to Wednesdays, 11:00-12:00. The location is the same, office 2.18 in E2.4. If none of the Wednesday time for the office hours and the Tuesdsay one (10:00 - 11:00) work for you, please let me know. Best,
Alberto |
Some referencesWritten on 31.10.23 by Alberto Dayan Dear students, the book
Tom M. Apostol: "Mathematical Analysis", second edition, Addison-Wesley Publishing Company (1981)
Contains a treatment of the Lebesgue integral. It also treats metric spaces and point-set topology, so it is also a good reference for you to refresh your… Read more Dear students, the book
Tom M. Apostol: "Mathematical Analysis", second edition, Addison-Wesley Publishing Company (1981)
Contains a treatment of the Lebesgue integral. It also treats metric spaces and point-set topology, so it is also a good reference for you to refresh your memory in case some of the prerequisites for this course are not very familiar.
See you on Thursday!
Alberto |
Functional Analysis I
Course Description
The main idea of functional analysis is that some problems in analysis can be solved by studying linear maps between vector spaces of functions. Such spaces are, in the vast majority of the cases, infinite dimensional, which means that linear functions between them can not be understood via finite matrices. Nonetheless, by endowing a vector space with a norm (i.e., an abstract notion of length of its elements), one has a metric (and hence a topology) to work with, and tools from analysis of metric spaces and topology generally apply. For instance, one can talk about continuity of linear maps between infinite dimensional vector spaces, and can define a notion of convergence of a sequence in a vector space. Therefore, throughout this course we will use ideas from linear algebra, analysis, and topology.
The topic that we will cover include some of the landmark results in classic functional analysis, such as:
- Normed spaces and linear operators between them
- Dual spaces and the Hahn-Banach theorem
- The open mapping theorem, the uniform boundedness principle and the closed graph theorem
- Weak * topologies and the Banach-Alaoglu theorem
- Hilbert spaces
- Spectral theory
Prerequisite
Linear algebra, metric spaces and a few basic notions from topology. Some knowledge from a first course of measure theory are recommended, though not required.
Language
The lectures will be given in English. The problem sessions will be given in English, unless the Assistant and all the students agree to have them in German.
Lectures
Tuesday and Thursday, 14:00 - 16:00 in HS IV, building E2.4
Problem Sessions
Wednesday, 12:00 (sharp/s.t.) - 14:00 in SR 5, building E2.4 (starting in November 8th)
Office Hours
Tuesday, 10:00 - 11:00 in room 1.01, building E2.4
Wednesday, 11:00 - 12:00 in room 2.18, building E2.4
Assignments and Exam
There will be weekly homework assignments. Students have the option of working on the problems in groups of two, maximum three people.
The exam will be oral, and in order to access it one has to obtain at least the half of the available points from the homework assignments.
References
W. Rudin: Functional Analysis, second edition, International series in Pure and Applied Mathematics, McGraw Hill
J. B. Conway: A Course in Functional Analysis, second edition, Graduate Texts in Mathematics 96, Springer
Contact
For any question regarding the course, don't hesitate to contact the instructor at dayan@math.uni-sb.de