Maria Infusino - Teaching
Winter Semester 2018
Topological Vector Spaces with
Patrick Michalski.
Lecture (2 hours per week): Tuesday 15.15 – 16.45, Room D301, by Dr Maria Infusino.
Fragestunde (1 hour per week): Wednesday 13.30 – 14.30, Room F408, by Dr Maria Infusino.
Tutorial (2 hours every two weeks):
Group A: Friday, 11:45 -13:30, Room P812, by Patrick Michalski.
Group B: Monday, 15.15 – 16.45, Room L829, by Patrick Michalski.
Contents
The aim of this course is to give an overview of the most important concepts and results
of the theory of topological vector spaces (TVS).
As the name suggests, this theory beautifully connects
topological and algebraic structures. The main focus
will be the study of TVS over the reals and particular
attention will be given to locally convex spaces. In the investigation of these
spaces we will restrict our attention to those questions which are of significance
for applications in real algebraic geometry.
For instance: the connection of locally convex spaces to seminorms,
the continuity of linear functionals on such spaces, the study of the finest locally convex topology,
approximation of positive polynomials by elements of quadratic modules,
integral representation of positive linear functionals, etc.
Prerequisites
The presentation will try to be as much self-contained
and systematic as possible, so the only prerequisite is a basic
knowledge of algebra and analysis. A prior exposure to functional analysis and some familiarity with general topology would be helpful, but not required.
Target group
BA, MA, LA (from 5.semester)
Validation
Spezialisierungsmodul im Schwerpunkt Reelle Geometrie und Algebra
Wahlmodul MA Mathematik
Wahlmodul MA LA
Wahlmodul/Spezielles Gebiet GymPO 2009
Exam
The final exam will be oral and to be scheduled individually. To enroll for the exam please contact Frau Gisela Cassola (Room F439).
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
in order to discuss 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. Moreover, the lecturer is available every week on
Wednesday 13:30-14:30, Room F404) for individual
meetings to discuss any question, problem and comment related to the course (
Fragestunde).
Tutorial Calendar
| Group A | Group B |
09.11 | 5.11 | Recap discussion about topological preliminaries |
16.11 | 19.11 | Discussion of solutions to Exercise Sheet 1 |
30.11 | 03.12 | Discussion of solutions to Exercise Sheet 2 |
14.12 | 17.12 | Discussion of solutions to Exercise Sheet 3 |
11.01 | 14.01 | Discussion of solutions to Christmas Assignment |
25.01 | 28.01 | Discussion of solutions to Exercise Sheet 5 |
12.02 | 12.02 | Discussion of solutions to Exercise Sheet 6 |
Lecture Notes
Lecture 1: Introduction to the course and Preliminaries on topological spaces
(last update on 25.10)
Lecture 2: More preliminaries on topological spaces
(last update on 30.10)
Lecture 3: Definition and main properties of a topological vector space
(last update on 6.11)
Lecture 4: Characterization of the filter of neighborhoods of the origin in a t.v.s. and Hausdorff t.v.s.
(last update on 21.11)
Lecture 5: Quotient of a t.v.s. and continuous linear mappings between t.v.s.
(last update on 21.11)
Lecture 6: Completeness for t.v.s.
(last update on 28.11)
Lecture 7: Finite dimensional t.v.s.
(last update on 05.12)
Lecture 8: Locally convex t.v.s.: definition by neighbourhoods
(last update on 11.12)
Lecture 9: Locally convex t.v.s.: connection to seminorms
(last update on 18.12)
Lecture 10: Locally convex t.v.s.: connection to seminorms and Hausdorffness
(last update on 8.01)
Lecture 11: Locally convex t.v.s.: finest locally convex topology and continuous linear mappings between locally convex spaces
(last update on 15.01)
Lecture 12: Hahn-Banach theorem
(last update on 5.02)
Lecture 13: Applications of Hahn-Banach theorem
(last update on 5.02)
Lecture 14: Applications of Hahn-Banach theorem
(last update on 8.02)
Lecture Notes (unique pdf file) (last update on 8.02)
Bibliography (last update on 8.02)
Exercise Sheets
Exercise Sheet 1 (due by 13.11.18)
Exercise Sheet 2 (due by 27.11.18)
Exercise Sheet 3 (due by 11.12.18)
Christmas Assignment (due by 8.01.19)
Exercise Sheet 5 (due by 22.01.19)
Exercise Sheet 6 (due by 5.02.19)
Recap Sheets
Recap Sheet 1 (handed out on 30.10.18)
Recap Sheet 2 (handed out on 13.11.18)
Recap Sheet 3 (handed out on 27.11.18)
Recap Sheet 4 (handed out on 11.12.18)
Recap Sheet 5 (handed out on 8.01.19)
Recap Sheet 6 (handed out on 22.01.19)
Further References
G. Köthe, Topological vector spaces I, Die Grundlehren der mathematischen Wissenschaften, 159, New York: Springer-Verlag, 1969. (available also in German)
M. Marshall, Positive Polynomials and Sums of Squares, 146, Math. Surveys & Monographs, AMS, 2008.
W. Rudin, Functional Analysis, second edition, McGraw-Hill Co, 1991.
H.H. Schaefer, M. P. Wolff, Topological vector spaces, second edition, Graduate Texts in Mathematics, 3. Springer-Verlag, New York,1999.
F. Trèves, Topological Vector Spaces, distributions, and kernels, Academic Press, 1967.
Announcements
The lecture on Tuesday 29.01.2019 is cancelled and will be recovered on Friday 08.02.2019, 11:45-13:15, Room P812
The Fragestunde of Wednesday 30.01.2019 is cancelled and replaced with a Fragestunde on Monday 04.02.2019, 15:00-16:00
The final oral exam will take place on Thursday 7.03.2019 and Thursday 11.04.2019
The Fragestunde of Monday 04.02.2019, 15:00-16:00 is moved to 14:00-15:00 in F410!
NEW: To officially enroll for the final exam, please contact Frau Gisela Cassola (Room F439) before 13.02.2019 .
Last update: 8.02.2019
