Course on Topological Vector Spaces

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

• Initial topology, topological vector spaces, weak topology.- Convexity, separation theorems, locally convex spaces.- Polars, bipolar theorem, polar topologies.- The theorems of Tikhonov and Alaoglu-Bourbaki.- The theorem of Mackey-Arens.- Topologies on E'', quasi-barrelled and barrelled spaces.- Reflexivity.- Completeness.- Locally convex final topology, topology of D(\Omega).- Precompact -- compact – complete.- The theorems of Banach--Dieudonne and Krein—Smulian.- The theorems of Eberlein--Grothendieck and Eberlein—Smulian.- The theorem of Krein.- Weakly compact sets in L_1(\mu).- \cB_0''=\cB.- The theorem of Krein—Milman.- A The theorem of Hahn-Banach.- B Baire's theorem and the uniform boundedness theorem.

“The material of the book is very carefully developed and even includes an introduction into the basics of topological and metric spaces. … At the beginning of each chapter, a brief outline of the subjects treated therein is given, while at the end, notes, comments and suggestions for further reading are included. … The book ends with an extensive reference list, an index and a very helpful index of notations.” (Wolfgang Lusky, Mathematical Reviews, December, 2021)“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)