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Research Group Real Algebraic Geometry > Prof. Dr. S. Kuhlmann > Mitarbeiter > Dr. Maria Infusino

Maria Infusino - Teaching

Winter semester 2017-18

Topological Vector Spaces II
with Patrick Michalski.

Lecture (2 hours per week): Friday 10.00 – 11.30, Room F426
Tutorial (2 hours every two weeks): Wednesday 13:30-15:00, Room D404

Contents
This course is meant to be a natural follow-up of the course "Topological Vector Spaces'' from the SS 2017. 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. A special 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 in 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 SS 2017. Therefore, a basic knowledge of the theory of t.v.s. is assumed. However, since the main focus of this course is on examples and applications, 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, Diplom (from 7.semester).

Validation
  • Spezialisierungsmodul MA Mathematik
  • Wahlmodul MA Mathematik
  • Wahlmodul MA LA
  • Wahlmodul/Spezielles Gebiet GymPO 2009

    • Language
    English

    Tutorial and Weekly Office hours
    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 (Wednesday 13:30-15:00, Room D404) 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 (Thursday 14:00-15:00, Room F408) for individual meetings to discuss any problem, comment and/or question related to the course.

    References Lecture Notes
    Lecture 1: Introduction to the course and Metrizable t.v.s. (last update on 27.10.2017)
    Lecture 2: More properties of metrizable t.v.s. and Fréchet spaces. (last update on 3.11.2017)
    Lecture 3: Examples of Fréchet spaces: the space of smooth functions and the Schwarz space. (last update on 10.11)
    Lecture 4: Inductive topologies and LF-spaces. (last update on 17.11.2017)
    Lecture 5: Properties of LF-spaces and examples. (last update on 24.11.2017)
    Lecture 6: Further example of LF-spaces, projective topologies and projective limits. (last update 08.12.2017)
    Lecture 7: Approximation procedures in function spaces. (last update 08.12.2017)
    Lecture 8: Approximation procedures in function spaces and some preliminaries on compactness (last update 14.12.2017)
    Lecture 9: Bounded subsets of a t.v.s. (last update 16.12.2017)
    Lecture 10: Bounded subsets of special classes of t.v.s. and the notion of polar (last update 12.01.2018)
    Lecture 11: Topologies on the dual of a t.v.s. (last update 23.01.2018)
    Lecture 12: Topologies on the dual of a t.v.s. and Banach-Alaoglu-Bourbaki Thm (last update 2.02.2018)
    Lecture 13: Tensor products of vector spaces (last update 2.02.2018)
    Lecture 14: Topologies on tensor products of l.c. t.v.s. (first part) (last update 9.02.2018)
    Lecture 15: Topologies on tensor products of l.c. t.v.s. (second part) (last update 16.02.2018)


    Lecture Notes (unique pdf file) (last update on 16.02.18)


    Problem Sheets
    Exercise Sheet 1 (to be handed in by 10.11.2017)
    Exercise Sheet 2 (to be handed in by 24.11.2017)
    Exercise Sheet 3 (to be handed in by 20.12.2017)
    Christmas Assignment (to be handed in by 08.01.2018)
    Exercise Sheet 5 (to be handed in by 19.01.2018)
    Exercise Sheet 6 (to be handed in by 02.02.2018)
    Exercise Sheet 7 (to be handed in by 16.02.2018)
    Solution to Exercise Sheet 7

    Recap Sheets
    Recap Sheet 1
    Recap Sheet 2
    Recap Sheet 3
    Recap Sheet 4
    Recap Sheet 5
    Recap Sheet 6


    Announcements




    Last update: 22.02.2018