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

Maria Infusino - Teaching

Winter semester 2015/16

Topological Vector Spaces


2 hours, Wednesday: 10.00 – 11.30, Room D404

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 (e.g. normed, seminormed and nuclear spaces). In the investigation of these spaces we will restrict our attention to those questions which are of significance for applications to the moment problem. For instance, the interplay among the following topics will be presented: the relation between locally convex spaces and seminorms, the finest locally convex topology on finite and infinite dimensional real vector spaces, approximation of positive polynomials by elements of quadratic modules, the study of closure of quadratic modules w.r.t. different topologies on real algebras, integral representation of positive linear functionals on function spaces (e.g. Riesz's representation theorem, Haviland's Theorem for both the full and the truncated moment problem), application of Putinar's Positivstellensatz to locally multiplicatively convex topological real algebras, moment problem for continuous functionals on the symmetric algebra of a locally convex space.

Target group
BA, MA, LA, Diplom (from 5.semester)
This course can be taken also as Master Wahlmodul.

Prerequisites
The course will include an introductory review of the fundamental notions in general topology which are needed to study TVS. Therefore, the only preliminary knowledge required for the course is classical analysis and basic linear algebra. A prior exposure to functional analysis would be helpful but not required.

References Language
English

Personal Tutorial
A weekly problem sheet will be distributed to allow the participants both to self-assess their progress and to explicitly work out more details of some of the results proposed in the lectures. A personal tutorial of one hour per week (Tuesday 3-4 pm) is offered to all the participants who will feel the need to discuss such exercises and/or further questions with the instructor.

Lecture Notes
Contents
Lecture 1: Preliminaries on topological spaces I (last update on 4.11)
Lecture 2: Preliminaries on topological spaces II (last update on 14.11)
Lecture 3: Definition and main properties of a topological vector space (last update on 14.11)
Lecture 4: Hausdorff topological vector spaces (last update on 18.11)
Lecture 5: Quotient t.v.s. and continuous linear mappings between t.v.s. (last update on 25.11)
Lecture 6: Completeness for t.v.s. (last update on 9.12)
Lecture 7: Theorem on the completion of a t.v.s. (last update on 29.01)
Lecture 8: Finite dimensional t.v.s. (last update on 29.01)
Lecture 9: Locally convex t.v.s.: definition by nbhoods (last update on 29.01)
Lecture 10: Locally convex t.v.s.: connection to seminorms (last update on 29.01)
Lecture 11: Locally convex t.v.s.: connection to seminorms and Hausdorfness (last update on 04.02)
Lecture 12: Locally convex t.v.s.: examples and finest locally convex topology (last update on 04.02)
Lecture 13: Continuous linear mappings between locally convex spaces and Hahn-Banach theorem (last update on 07.03)
Lecture 14: Applications of Hahn-Banach theorem (last update on 29.07)

Lecture Notes (unique pdf file) (last update on 29.07)

Problem Sheets
Sheet 1
Sheet 2
Sheet 3
Sheet 4
Sheet 5
Sheet 6
Sheet 7
Christmas assignment
Sheet 9
Sheet 10

Announcements




Last update: 29.07.2016