© 2020

A Course on Topological Vector Spaces


Part of the Compact Textbooks in Mathematics book series (CTM)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Jürgen Voigt
    Pages 23-28
  3. Jürgen Voigt
    Pages 37-43
  4. Jürgen Voigt
    Pages 53-62
  5. Jürgen Voigt
    Pages 63-69
  6. Jürgen Voigt
    Pages 71-79
  7. Jürgen Voigt
    Pages 93-95
  8. Jürgen Voigt
    Pages 113-117
  9. Jürgen Voigt
    Pages 119-124
  10. Jürgen Voigt
    Pages 125-130
  11. Jürgen Voigt
    Pages 131-138
  12. Back Matter
    Pages 139-155

About this book


This book provides an introduction to the theory of topological vector spaces, with a focus on locally convex spaces. It discusses topologies in dual pairs, culminating in the Mackey-Arens theorem, and also examines the properties of the weak topology on Banach spaces, for instance Banach’s theorem on weak*-closed subspaces on the dual of a Banach space (alias the Krein-Smulian theorem), the Eberlein-Smulian theorem, Krein’s theorem on the closed convex hull of weakly compact sets in a Banach space, and the Dunford-Pettis theorem characterising weak compactness in L1-spaces. Lastly, it addresses topics such as the locally convex final topology, with the application to test functions D(Ω) and the space of distributions, and the Krein-Milman theorem. 

The book adopts an “economic” approach to interesting topics, and avoids exploring all the arising side topics. Written in a concise mathematical style, it is intended primarily for advanced graduate students with a background in elementary functional analysis, but is also useful as a reference text for established mathematicians. 


topology convex spaces polars bipolar theorem Alaoglu-Bourbaki Tikhonov Mackey-Arens reflexivity Krein

Authors and affiliations

  1. 1.Institut für AnalysisTechnische Universität DresdenDresdenGermany

About the authors

Jürgen Voigt is Professor at the Institute of Analysis of the Technische Universität in Dresden, Germany.

Bibliographic information

  • Book Title A Course on Topological Vector Spaces
  • Authors Jürgen Voigt
  • Series Title Compact Textbooks in Mathematics
  • Series Abbreviated Title Compact Textbooks in Mathematics
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-030-32944-0
  • eBook ISBN 978-3-030-32945-7
  • Series ISSN 2296-4568
  • Series E-ISSN 2296-455X
  • Edition Number 1
  • Number of Pages VIII, 155
  • Number of Illustrations 0 b/w illustrations, 1 illustrations in colour
  • Topics Functional Analysis
  • Buy this book on publisher's site
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“The book may be highly recommended to all students and researchers with some knowledge of Banach or Hilbert space oriented functional analysis who want to learn its general abstract foundations.” (Jochen Wengenroth, zbMATH 1453.46001, 2021)