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

Maria Infusino - Teaching

Summer Semester 2017

Topological Vector Spaces
with Patrick Michalski

Lecture (2 hours per week): Friday 10.00 – 11.30, Room F426
Tutorial (2 hours per week): Tuesday 13:30-15:00, 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. 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 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.

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 basic analysis and basic linear algebra. A prior exposure to functional analysis would be helpful but not required.

Target group
BA (from 4.semester), MA, LA.

  • Ergänzungsmodul BA Mathematik
  • Wahlmodul MA Mathematik
  • Wahlmodul MA LA
  • Wahlmodul/Spezielles Gebiet GymPO 2009

  • Language

    Weekly Tutorial and Office hours
    A weekly problem sheet will be distributed 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 week (Tuesday 13:30-15:00, Room D404) to all the participants in order to discuss their solutions to the problem sheets and/or further questions. 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.

    Lecture Notes
    Contents (last update on 29.07)
    Lecture 1: Introduction and Preliminaries on topological spaces (last update on 04.05)
    Lecture 2: More preliminaries on topological spaces (last update on 05.05)
    Lecture 3: Definition and main properties of a topological vector space (last update on 12.05)
    Lecture 4: More properties of a t.v.s. and Hausdorff t.v.s. (last update on 19.05)
    Lecture 5: Quotient t.v.s. and continuous linear mappings between t.v.s. (last update on 26.05)
    Lecture 6: Completeness for t.v.s. (last update on 2.06)
    Lecture 7: Finite dimensional t.v.s. (last update on 13.06)
    Lecture 8: Locally convex t.v.s.: definition by neighbourhoods (last update on 25.06)
    Lecture 9: Locally convex t.v.s.: connection to seminorms (last update on 27.06)
    Lecture 10: Locally convex t.v.s.: connection to seminorms and Hausdorfness (last update on 9.07)
    Lecture 11: Locally convex t.v.s.: Hausdorf locally convex t.v.s. and finest locally convex topology (last update on 11.07)
    Lecture 12: Finite topology and continuous linear mappings between locally convex spaces (last update on 19.07)
    Lecture 13: Hahn-Banach theorem and its applications (last update on 19.07)
    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 (due on May 5th)
    Sheet 2 (due on May 12th)
    Sheet 3 (due on May 19th)
    Sheet 4 (due on May 26th)
    Sheet 5 (due on June 2nd)
    Sheet 6 (due on June 13th)
    Sheet 7 (due on June 23rd)
    Sheet 8 (due on June 30th)
    Sheet 9 (due on June 7th)
    Sheet 10(due on July 18th)
    Bonus Sheet (due on July 25th, see instructions)

    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 Friday 12.05.2017 will be held in Room D406 instead that the usual room!
  • The office hour of Thursday 8.06.2017 is cancelled. Please, contact me via email to take an alternative appointment..
  • The lecture on Friday 9.06.2017 is moved to Tuesday 13.06.2017, 13:30-15:00, Room D404 instead that the usual tutorial!
  • The lecture on Friday 16.06.2017 is cancelled and will be recovered on Monday 26.06.2017, 10:00-11:30, Room M801
  • The tutorial on Tuesday 11.07.2017 is moved to Friday 7.07.2017, 10:00-11:30, Room F426
  • The lecture on Friday 7.07.2017 is moved to Tuesday 11.07.2017, 13:30-15:00, Room D404
  • To officially enroll for the final oral exam please contact Frau Gisela Cassola (Room F439) before Friday 21st.

    Last update: 29.07.2017