A Course on Topological Vector Spaces

  • Jürgen Voigt

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

  • Jürgen Voigt
    • 1
  1. 1.Institut für AnalysisTechnische Universität DresdenDresdenGermany

Bibliographic information

  • DOI
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-32944-0
  • Online ISBN 978-3-030-32945-7
  • Series Print ISSN 2296-4568
  • Series Online ISSN 2296-455X
  • Buy this book on publisher's site
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