Maria Infusino - Teaching
Winter Semester 2019/2020
Topological Vector Spaces II with
Patrick Michalski.
Lecture (2 hours per week): Wednesdays 13:30-15:00, Room M631 by Dr Maria Infusino.
Tutorial (2 hours every two weeks): Thursdays 13:30-15:00, Room D406 by Patrick Michalski.
Fragestunden (2 hours per week): Thursdays 11:45-13:15, Room F408 by Dr Maria Infusino.
Contents
This course is meant to be a natural follow-up of the course "Topological Vector Spaces"
from the WS 2018/19. The main purpose is to develop some more advanced topics in the
theory of topological vector spaces (t.v.s.) with a particular focus on examples and
problems which show the power of the general results (introduced in the previous course)
in applications. Special classes of t.v.s., e.g. Frechét and LF-spaces, will be introduced
with an eye on their practical use in approximation techniques in function spaces.
Particular attention will be given to theory of duality and tensor products of t.v.s.
with an exposition of the various topologies naturally carried on such structures and
examples occurring in distribution theory. These structures also play an important role
in some problems connected to real algebraic geometry that will be outlined in this
course and could be a starting point for a master thesis in the research stream
investigated in the
Schwerpunkt Reelle Geometrie und Algebra.
Prerequisites
The course is a continuation of the course "Topological Vector Spaces"
from the WS 2018/19, so basic knowledge of the theory of t.v.s. is assumed.
However, participants with a prior exposure to functional analysis and topology might also be able to follow without having attended the previous course.
Target group
BA, MA, LA (from 7.semester)
Validation
Hauptmodul
Spezialisierungsmodul im Schwerpunkt Reelle Geometrie und Algebra
Wahlmodul MA Mathematik
Wahlmodul MA LA
Wahlmodul/Spezielles Gebiet GymPO 2009
Credits
5 ECTS
Exam
The final exam will be oral and to be scheduled individually. In order to be admitted at the exam, the participants need to achieve at least 50%
of the total number of points assigned in the Exercise Sheets as well as present a solution in the tutorial at least once.
Language
English
Exercise and Recap Sheets
An
exercise sheet will be distributed every two weeks to both assess the progress of the participants
and allow them to explicitly work out more details of some results proposed in the lectures.
A
tutorial is offered every two weeks to all the participants
for the discussion of their solutions to the exercise sheets.
During the weeks when there is no tutorial, a set of
recap questions
will be distributed to help the participants in self-assessing their learning process
in preparation for the oral exam.
Fragestunden
The lecturer is available every week on
Thursdays 11:45-13:15, Room F408) for individual
meetings to discuss any question, problem and comment related to the course.
Tutorial Calendar
| 19.11.2019 | Discussion of solutions to Exercise Sheet 1 |
03.12.2019 | Discussion of solutions to Exercise Sheet 2 |
19.12.2019 | Discussion of solutions to Exercise Sheet 3 |
16.01.2020 | Discussion of solutions to Exercise Sheet 4 |
30.01.2020 | Discussion of solutions to Exercise Sheet 5 |
13.02.2020 | Discussion of solutions to Exercise Sheet 6 and Bonus Sheet |
Lecture Notes
Lecture 0: Introduction to the course
(last update on 29.10.2019)
Lecture 1: Metrizable t.v.s.
(last update on 31.10.2019)
Lecture 2: More properties of metrizable t.v.s.
(last update on 7.11.2019)
Lecture 3: Fréchet spaces.
(last update on 14.11.2019)
Lecture 4: Inductive topologies and LF-spaces.
(last update on 20.11.2019)
Lecture 5: Properties of LF-spaces and examples.
(last update on 27.11.2019)
Lecture 6: Projective topologies, projective limits and an introduction to the open mapping theorem.
(last update 13.12.2019)
Lecture 7: Open Mapping Theorem and Closed Graph Theorem.
(last update 16.12.2019)
Lecture 8: Some preliminaries on compactness and introduction to bounded subsets of a t.v.s.
(last update 18.12.2019)
Lecture 9: More on bounded subsets of a t.v.s. and bounded linear maps.
(last update 08.01.2020)
Lecture 10: Bounded subsets of special classes of t.v.s. and the notion of polar topologies
(last update 22.01.2020)
Lecture 11: Properties of polar topologies
(last update 23.01.2020)
Lecture 12: Banach-Alaoglu-Bourbaki Theorem and Tensor products of vector spaces
(last update 5.02.2020)
Lecture 13: Properties of tensor products of vector spaces and the
$\pi$-topology
(last update 5.02.2020)
Lecture 14: More on the
$\pi-$topology and the
$\varepsilon-$topology
(last update 12.02.2020)
Lecture Notes (unique pdf file) (last update on 12.02.2020)
Bibliography (last update on 7.11.2019)
Exercise Sheets
Exercise Sheet 1 (due by 13.11.19)
Exercise Sheet 2 (due by 27.11.19)
Exercise Sheet 3 (due by 11.12.19)
Christmas Assignment (due by 8.01.20)
Exercise Sheet 5 (due by 22.01.20)
Exercise Sheet 6 (due by 5.02.20)
Bonus Sheet (due by 12.02.20)
Solution to Bonus Sheet (handed out on 14.02.20)
Interactive Sheets
Interactive Sheet 1 (worked out on 30.10.19)
Interactive Sheet 2 (Group 1) (worked out on 18.12.19)
Interactive Sheet 2 (Group 2) (worked out on 18.12.19)
Interactive Sheet 3 (worked out on 23.01.19)
Recap Sheets
Recap Sheet 0 (handed out on 30.10.19)
Recap Sheet 1 (handed out on 13.11.19)
Recap Sheet 2 (handed out on 27.11.19)
Recap Sheet 3 (handed out on 11.12.19)
Recap Sheet 4 (handed out on 8.01.20)
Recap Sheet 5 (handed out on 23.01.20)
Recap Sheet 6 (handed out on 12.02.20)
Announcements
The Fragestunde of Thursday 19.12.2019 is moved to Tuesday 17.12.2019, 10:00-11:30.
The lecture on Wednesday 15.01.2020 is cancelled and will be recovered on Thursday 23.01.2020, 13:30-15:00, Room D406
The Fragestunde of Thursday 16.01.2020 is is moved to Monday 13.01.2019, 10:00-11:30.
The final oral exams (validation as Wahlmodul) will take place on 10.03.2020 in F408 from 10:00.
NEW: To officially enroll for the final exam, please contact Frau Barjasic (Room F439).
Last update: 14.02.2020
